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Mathematics · Class 12 Science

Ch 9Applications of Integration — Class 12 Mathematics, concept-first.

One of the earliest mathematicians to compute areas and volumes of geometric shapes rigorously was Archimedes of Syracuse (287 BCE–212 BCE), a Greek mathematician, physicist, engineer and inventor.

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Definite Integral as the Limit of a Sum

Given a bounded function on a closed interval , partition into subintervals with . In each subinterval pick any point (with ) and form the Riemann sum As in such a way that the width of the widest subinterval , this sum…

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Chapter contents

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9.1

Introduction

One of the earliest mathematicians to compute areas and volumes of geometric shapes rigorously was Archimedes of Syracuse (287 BCE–212 BCE), a Greek mathematician, physicist, engineer and inventor.

9.2

Definite Integral as the Limit of a Sum

This section makes precise the informal picture used above — an area built by summing infinitely many thin strips — as the formal definition of the definite integral.

9.2.1

Riemann Integral

Setting up the Riemann sum. Let be a real-valued, bounded function on the closed interval , . Unlike in a first geometric picture, need not keep the same sign throughout — it may take both positive an…

+Exercise 9.1i3 questions
  1. Q1Find an approximate value of $\displaystyle\int_1^{1.5} x\,dx$ by applying the left-end rule with the partition $\{1.1,\ 1.2,\ 1.3,\ 1.4,\ 1…Free
  2. Q2Find an approximate value of $\displaystyle\int_1^{1.5} x^2\,dx$ by applying the right-end rule with the partition $\{1.1,\ 1.2,\ 1.3,\ 1.4,…Preview
  3. Q3Find an approximate value of $\displaystyle\int_1^{1.5} (2-x)\,dx$ by applying the mid-point rule with the partition $\{1.1,\ 1.2,\ 1.3,\ 1.…Preview
9.2.2

Limit Formula to Evaluate a Definite Integral

Rather than choose an arbitrary partition, divide into equal subintervals , so that Put ; then , .

+Exercise 9.2i1 question
  1. Q1Evaluate the following integrals as the limits of sums: (i) $\displaystyle\int_0^1 (5x+4)\,dx$ (ii) $\displaystyle\int_1^2 (4x^2-1)\,dx$Preview
9.3

Fundamental Theorems of Integral Calculus and their Applications

Evaluating as a limit of Riemann sums (§9.2) is correct but tedious, even for a very simple . Newton and Leibniz, working independently at around the same time, devised a vastly easier method based on…

+Exercise 9.3i2 questions
  1. Q1Evaluate the following definite integrals: (i) $\displaystyle\int_3^4 \dfrac{dx}{x^2-4}$ (ii) $\displaystyle\int_{-1}^1 \dfrac{dx}{x^2+2x+5}…Free
  2. Q2Evaluate the following integrals using properties of integration: (i) $\displaystyle\int_{-5}^5 x\cos\!\left(\dfrac{e^x-1}{e^x+1}\right)dx$…Preview
9.4

Bernoulli's Formula

The indefinite integral of a product becomes especially simple to finish in one pass — instead of repeated integration by parts — when is a polynomial and can be integrated over and over with ease (e.…

+Exercise 9.4i4 questions
  1. Q1$\displaystyle\int_0^1 x^3e^{-2x}\,dx$Free
  2. Q2$\displaystyle\int_0^1 \dfrac{\sin(3\tan^{-1}x)\tan^{-1}x}{1+x^2}\,dx$Free
  3. Q3$\displaystyle\int_0^{1/\sqrt2} \dfrac{e^{\sin^{-1}x}\sin^{-1}x}{\sqrt{1-x^2}}\,dx$Preview
  4. Q4$\displaystyle\int_0^{\pi/2} x^2\cos2x\,dx$Preview
9.5

Improper Integrals

The Riemann integral (§9.2.1) requires the interval to be finite and to be finite at every point of . Many physical applications instead need integrals of the shape where is real and is continuous on…

+Exercise 9.5i1 question
  1. Q1Evaluate the following: (i) $\displaystyle\int_0^{\pi/2}\dfrac{dx}{1+5\cos^2x}$ (ii) $\displaystyle\int_0^{\pi/2}\dfrac{dx}{5+4\sin^2x}$Preview
9.6

Reduction Formulae

Certain definite integrals with a repeated power (an index) can be evaluated by an index-reduction method instead of direct computation.

+Exercise 9.6i1 question
  1. Q1Evaluate the following: (i) $\displaystyle\int_0^{\pi/2}\sin^{10}x\,dx$ (ii) $\displaystyle\int_0^{\pi/2}\cos^7x\,dx$ (iii) $\displaystyle\i…Preview
9.7

Gamma Integral

This section studies a special improper integral of the form , where is a positive integer.

+Exercise 9.7i2 questions
  1. Q1Evaluate the following: (i) $\displaystyle\int_0^\infty x^5e^{-3x}\,dx$ (ii) $\displaystyle\int_0^{\pi/2}\dfrac{e^{-\tan x}}{\cos^6x}\,dx$Free
  2. Q2If $\displaystyle\int_0^\infty e^{-\alpha x^2}x^3\,dx=32,\ \alpha>0$, find $\alpha$.Preview
9.8

Evaluation of a Bounded Plane Area by Integration

Section 9.2.1's Remarks already connected the sign of to geometric area — a non-negative integrand yields area directly.

9.8.1

Area of the Region Bounded by a Curve, x-axis and the Lines x=a, x=b

Case (i): curve above the -axis. Let , , be a continuous curve lying entirely above the -axis (in the first or second quadrant) between and , so throughout.

9.8.2

Area of the Region Bounded by a Curve, y-axis and the Lines y=c, y=d

The mirror-image formulas for a curve related to the -axis, obtained by the same strip argument with the roles of and interchanged.

9.8.3

Area of the Region Bounded Between Two Curves

Case (i): between two curves, integrating in . Let and be curves with for all . The region bounded between them and the ordinates is divided into thin vertical strips of width and height .

+Exercise 9.8i10 questions
  1. Q1Find the area of the region bounded by $3x-2y+6=0$, $x=-3$, $x=1$ and $x$-axis.Free
  2. Q2Find the area of the region bounded by $2x-y+1=0$, $y=-1$, $y=3$ and $y$-axis.Free
  3. Q3Find the area of the region bounded by the curve $2+x-x^2+y=0$, $x$-axis, $x=-3$ and $x=3$.Free
  4. Q4Find the area of the region bounded by the line $y=2x+5$ and the parabola $y=x^2-2x$.Preview
  5. Q5Find the area of the region bounded between the curves $y=\sin x$ and $y=\cos x$ and the lines $x=0$ and $x=\pi$.Preview
  6. Q6Find the area of the region bounded by $y=\tan x$, $y=\cot x$ and the lines $x=0$, $x=\dfrac{\pi}{2}$, $y=0$.Preview
  7. Q7Find the area of the region bounded by the parabola $y^2=x$ and the line $y=x-2$.Preview
  8. Q8Father of a family wishes to divide his square field bounded by $x=0$, $x=4$, $y=4$ and $y=0$ along the curve $y^2=4x$ and $x^2=4y$ into thr…Preview
  9. Q9The curve $y=(x-2)^2+1$ has a minimum point at $P$. A point $Q$ on the curve is such that the slope of $PQ$ is 2. Find the area bounded by t…Preview
  10. Q10Find the area of the region common to the circle $x^2+y^2=16$ and the parabola $y^2=6x$.Preview
9.9

Volume of a Solid Obtained by Revolving Area About an Axis

Solids of revolution. When a plane region is given one complete rotation ( radians) about a fixed axis lying in its own plane, it sweeps out a solid of revolution.

+Exercise 9.9i6 questions
  1. Q1Find, by integration, the volume of the solid generated by revolving about the $x$-axis, the region enclosed by $y=2x^2$, $y=0$ and $x=1$.Free
  2. Q2Find, by integration, the volume of the solid generated by revolving about the $x$-axis, the region enclosed by $y=e^{-2x}$, $y=0$, $x=0$ an…Free
  3. Q3Find, by integration, the volume of the solid generated by revolving about the $y$-axis, the region enclosed by $x^2=1+y$ and $y=3$.Preview
  4. Q4The region enclosed between the graphs of $y=x$ and $y=x^2$ is denoted by $R$. Find the volume generated when $R$ is rotated through $360^\c…Preview
  5. Q5Find, by integration, the volume of the container which is in the shape of a right circular conical frustum of height $2\text{ m}$, whose tw…Preview
  6. Q6A watermelon has an ellipsoid shape which can be obtained by revolving an ellipse with major-axis 20 cm and minor-axis 10 cm about its major…Preview
9.10

Choose the Correct or the Most Suitable Answer

This is the chapter's closing 20-question multiple-choice review (Exercise 9.10, "choose the correct or the most suitable answer"), drawing on every tool built across the chapter — the limit-of-a-sum…

+Exercise 9.10i20 questions
  1. Q1The value of $\displaystyle\int_0^{2/3}\dfrac{dx}{\sqrt{4-9x^2}}$ is (1) $\dfrac{\pi}{6}$ (2) $\dfrac{\pi}{2}$ (3) $\dfrac{\pi}{4}$ (4) $\pi…Free
  2. Q2The value of $\displaystyle\int_{-1}^2 |x|\,dx$ is (1) $\dfrac12$ (2) $\dfrac32$ (3) $\dfrac52$ (4) $\dfrac72$Free
  3. Q3For any value of $n\in\mathbb{Z}$, $\displaystyle\int_0^\pi e^{\cos^2x}\cos^3\big[(2n+1)x\big]\,dx$ is (1) $\dfrac{\pi}{2}$ (2) $\pi$ (3) $0…Free
  4. Q4The value of $\displaystyle\int_{-\pi/2}^{\pi/2}\sin^2x\cos x\,dx$ is (1) $\dfrac32$ (2) $\dfrac12$ (3) $0$ (4) $\dfrac23$Preview
  5. Q5The value of $\displaystyle\int_{-4}^4\left[\tan^{-1}\!\left(\dfrac{x^2}{x^4+1}\right)+\tan^{-1}\!\left(\dfrac{x^4+1}{x^2}\right)\right]dx$…Preview
  6. Q6The value of $\displaystyle\int_{-\pi/4}^{\pi/4}\left(\dfrac{2x^7-3x^5+7x^3-x+1}{\cos^2x}\right)dx$ is (1) $4$ (2) $3$ (3) $2$ (4) $0$Preview
  7. Q7If $f(x)=\displaystyle\int_0^x t\cos t\,dt$, then $\dfrac{df}{dx}=$ (1) $\cos x-x\sin x$ (2) $\sin x+x\cos x$ (3) $x\cos x$ (4) $x\sin x$Preview
  8. Q8The area between $y^2=4x$ and its latus rectum is (1) $\dfrac23$ (2) $\dfrac43$ (3) $\dfrac83$ (4) $\dfrac53$Preview
  9. Q9The value of $\displaystyle\int_0^1 x(1-x)^{99}\,dx$ is (1) $\dfrac{1}{11000}$ (2) $\dfrac{1}{10100}$ (3) $\dfrac{1}{10010}$ (4) $\dfrac{1}{…Preview
  10. Q10The value of $\displaystyle\int_0^\pi \dfrac{dx}{1+5^{\cos x}}$ is (1) $\dfrac{\pi}{2}$ (2) $\pi$ (3) $\dfrac{3\pi}{2}$ (4) $2\pi$Preview
  11. Q11If $\dfrac{\Gamma(n+2)}{\Gamma(n)}=90$ then $n$ is (1) $10$ (2) $5$ (3) $8$ (4) $9$Preview
  12. Q12The value of $\displaystyle\int_0^{\pi/6}\cos^3 3x\,dx$ is (1) $\dfrac23$ (2) $\dfrac29$ (3) $\dfrac19$ (4) $\dfrac13$Preview
  13. Q13The value of $\displaystyle\int_0^\pi \sin^4x\,dx$ is (1) $\dfrac{3\pi}{10}$ (2) $\dfrac{3\pi}{8}$ (3) $\dfrac{3\pi}{4}$ (4) $\dfrac{3\pi}{2…Preview
  14. Q14The value of $\displaystyle\int_0^\infty e^{-3x}x^2\,dx$ is (1) $\dfrac{7}{27}$ (2) $\dfrac{5}{27}$ (3) $\dfrac{4}{27}$ (4) $\dfrac{2}{27}$Preview
  15. Q15If $\displaystyle\int_0^a \dfrac{1}{4+x^2}\,dx=\dfrac{\pi}{8}$ then $a$ is (1) $4$ (2) $1$ (3) $3$ (4) $2$Preview
  16. Q16The volume of solid of revolution of the region bounded by $y^2=x(a-x)$ about $x$-axis is (1) $\pi a^3$ (2) $\dfrac{\pi a^3}{4}$ (3) $\dfrac…Preview
  17. Q17If $f(x)=\displaystyle\int_1^x \dfrac{e^{\sin u}}{u}\,du,\ x>1$ and $\displaystyle\int_1^3 \dfrac{e^{\sin x^2}}{x}\,dx=\dfrac12\big[f(a)-f(1…Preview
  18. Q18The value of $\displaystyle\int_0^1 \left(\sin^{-1}x\right)^2\,dx$ is (1) $\dfrac{\pi^2}{4}-1$ (2) $\dfrac{\pi^2}{4}+2$ (3) $\dfrac{\pi^2}{4…Preview
  19. Q19The value of $\displaystyle\int_0^a \left(\sqrt{a^2-x^2}\right)^3\,dx$ is (1) $\dfrac{\pi a^3}{16}$ (2) $\dfrac{3\pi a^4}{16}$ (3) $\dfrac{3…Preview
  20. Q20If $\displaystyle\int_0^x f(t)\,dt = x+\int_x^1 tf(t)\,dt$, then the value of $f(1)$ is (1) $\dfrac12$ (2) $2$ (3) $1$ (4) $\dfrac34$Preview
9.11

Summary

(1) Definite integral as the limit of a sum. (i) (ii) On :

Sample & Board Papers

Sample papers and previous-year board questions for this subject.

+Show 46 questions46 questions
  1. Q1The surface area of the solid of revolution of the region bounded by $y=2x$, $x=0$ and $x=2$ about $x$-axis is : (a) $8\sqrt{5}\pi$ (b) $2\s…Preview
  2. Q2The area of the region bounded by the graphs of $y=\sin x$ and $y=\cos x$ between $x=0$ and $x=\dfrac{\pi}{4}$ is : (a) $\sqrt{2}+1$ (b) $\s…Preview
  3. Q3$\displaystyle\int_0^{2a} f(x)\,dx = 2\int_0^{a} f(x)\,dx$ if : (a) $f(2a-x)=f(x)$ (b) $f(a-x)=f(x)$ (c) $f(x)=-f(x)$ (d) $f(-x)=f(x)$Preview
  4. Q4The volume generated by rotating the triangle with vertices at $(0,0)$, $(3,0)$ and $(3,3)$ about $x$-axis is : (a) $18\pi$ (b) $2\pi$ (c) $…Preview
  5. Q5Evaluate : $\displaystyle\int \sin^6 x\, dx$.Preview
  6. Q6Find the surface area of the solid generated by revolving one arc of the cycloid $x=a(t+\sin t)$, $y=a(1+\cos t)$ about its base ($x$-axis).Preview
  7. Q7Find the area of the region bounded by the ellipse $\dfrac{x^2}{9}+\dfrac{y^2}{5}=1$ between the two latus rectums.Preview
  8. Q8Volume of the solid obtained by revolving the area of the ellipse $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$ about major and minor axes are i…Preview
  9. Q9The value of $\displaystyle\int_0^{\pi/4} \cos^3 2x\, dx$ is : (a) $\dfrac{2}{3}$ (b) $\dfrac{1}{3}$ (c) 0 (d) $\dfrac{2\pi}{3}$Preview
  10. Q10If $I_n = \displaystyle\int \cos^n x\, dx$ then $I_n =$ (a) $\dfrac{-1}{n}\cos^{n-1}x\sin x + \left(\dfrac{n-1}{n}\right)I_{n-2}$ (b) $\cos^…Preview
  11. Q11The value of $\displaystyle\int_0^{\pi/2} \dfrac{\sin x - \cos x}{1 + \sin x \cos x}\, dx$ is : (a) $\dfrac{\pi}{2}$ (b) 0 (c) $\dfrac{\pi}{…Preview
  12. Q12Find the area of the region bounded by the ellipse $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$, by integration.Preview
  13. Q13Find the length of the curve $x = a(t - \sin t)$, $y = a(1 - \cos t)$ between $t = 0$ and $t = \pi$.Preview
  14. Q14The value of $\int_0^{\pi} \sin^4 x \, dx$ is : (a) $0$ (b) $\dfrac{3\pi}{16}$ (c) $\dfrac{3\pi}{8}$ (d) $\dfrac{3}{16}$Preview
  15. Q15The volume of the solid generated by rotating the triangle with vertices at $(0, 0)$, $(3, 0)$ and $(3, 3)$ about $x$-axis is : (a) $36\pi$…Preview
  16. Q16The surface area of the solid obtained by revolving the region bounded by $y = 2x$, $x = 0$ and $x = 2$ about $x$-axis, is : (a) $\sqrt{5}\p…Preview
  17. Q17$\displaystyle\int_0^{\infty} x^6 e^{-\frac{x}{2}} \, dx = $ (a) $2^6 \cdot 6!$ (b) $\dfrac{6!}{2^7}$ (c) $2^7 \cdot 6!$ (d) $\dfrac{6!}{2^6…Preview
  18. Q18Evaluate: $\displaystyle\int \cos^5 x \, dx$Preview
  19. Q19Find the common area enclosed by the parabolas $4y^2 = 9x$; $3x^2 = 16y$.Preview
  20. Q20Find the length of the curve $\left(\dfrac{x}{a}\right)^{2/3} + \left(\dfrac{y}{a}\right)^{2/3} = 1$.Preview
  21. Q21The value of $\displaystyle\int_{0}^{\pi/2} \dfrac{\tan x - \cot x}{1 + \tan x \cot x} \, dx$ is : (a) $\dfrac{\pi}{4}$ (b) $\pi$ (c) $\dfra…Preview
  22. Q22The surface area of the solid of revolution of the region bounded by $x^2 + y^2 = 4$, $x = -2$ and $x = 2$ about $x$-axis is : (a) $64\pi$ (…Preview
  23. Q23Prove that $\displaystyle\int_{\pi/6}^{\pi/3} \dfrac{dx}{1+\sqrt{\cot x}} = \displaystyle\int_{\pi/6}^{\pi/3} \dfrac{dx}{1+\sqrt{\tan x}}$.Preview
  24. Q24Derive the formula for the volume of a cylinder with radius 'r' and height 'h' by using integration.Preview
  25. Q25The value of $\displaystyle\int_{0}^{\pi} \sin^4 x\, dx$ is : (a) $\dfrac{3\pi}{2}$ (b) $\dfrac{3\pi}{10}$ (c) $\dfrac{3\pi}{8}$ (d) $\dfrac…Preview
  26. Q26The value of $\displaystyle\int_{0}^{2/3}\dfrac{dx}{\sqrt{4-9x^2}}$ is : (a) $\pi$ (b) $\dfrac{\pi}{6}$ (c) $\dfrac{\pi}{2}$ (d) $\dfrac{\pi…Preview
  27. Q27Prove that $\displaystyle\int_{0}^{\pi/2}\dfrac{f(\sin x)}{f(\sin x)+f(\cos x)}\,dx=\dfrac{\pi}{4}$.Preview
  28. Q28The value of $\displaystyle\int_{0}^{1}x(1-x)^{99}\,dx$ is : (a) $\dfrac{1}{10010}$ (b) $\dfrac{1}{11000}$ (c) $\dfrac{1}{10001}$ (d) $\dfra…Preview
  29. Q29The value of $\displaystyle\int_{0}^{\infty}e^{-3x}x^2\,dx$ is : (a) $\dfrac{4}{27}$ (b) $\dfrac{7}{27}$ (c) $\dfrac{2}{27}$ (d) $\dfrac{5}{…Preview
  30. Q30Show that $\displaystyle\int_{0}^{\pi/3}\dfrac{\sec x\tan x}{1+\sec^2x}\,dx=\tan^{-1}(2)-\dfrac{\pi}{4}$.Preview
  31. Q31The area between $y^2=4x$ and its latus rectum is : (a) $\dfrac83$ (b) $\dfrac23$ (c) $\dfrac53$ (d) $\dfrac43$Preview
  32. Q32The value of $\displaystyle\int_{0}^{\pi/3}\tan x\,dx$ is : (a) $-\log 2$ (b) $\log 2$ (c) $-\log 3$ (d) $\log 3$Preview
  33. Q33Evaluate : $\displaystyle\int_{b}^{\infty}\dfrac{1}{a^2+x^2}\,dx$, $a>0$, $b\in\mathbb{R}$.Preview
  34. Q34The area between $y^2=4x$ and its latus rectum is : (a) $\dfrac83$ (b) $\dfrac23$ (c) $\dfrac53$ (d) $\dfrac43$Preview
  35. Q35The value of $\displaystyle\int_{0}^{a}\left(\sqrt{a^2-x^2}\right)^3\,dx$ is : (a) $\dfrac{3\pi a^2}{8}$ (b) $\dfrac{\pi a^3}{16}$ (c) $\dfr…Preview
  36. Q36Evaluate : $\displaystyle\int_{0}^{\pi/2}\sin^{10}x\,dx$Preview
  37. Q37Evaluate : $\displaystyle\int_{\pi/8}^{3\pi/8}\dfrac{1}{1+\sqrt{\tan x}}\,dx$Preview
  38. Q38(a) Find the area of the region bounded by the ellipse $\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1$ **OR** (b) Find the vertex, focus and equation…Preview
  39. Q39The value of $\displaystyle\int_{0}^{\frac23}\dfrac{dx}{\sqrt{4-9x^2}}$ is : (a) $\dfrac{\pi}{4}$ (b) $\dfrac{\pi}{6}$ (c) $\pi$ (d) $\dfrac…Preview
  40. Q40The volume of solid of revolution of the region bounded by $y^2=x(a-x)$ about $x$-axis is : (a) $\dfrac{\pi a^3}{5}$ (b) $\pi a^3$ (c) $\dfr…Preview
  41. Q41If $f(x)=\sin x$, then prove that $\displaystyle\int_{0}^{\pi}f(x)\,dx=2\int_{0}^{\frac{\pi}{2}}f(x)\,dx$Preview
  42. Q42Evaluate $\displaystyle\int_{0}^{\frac{\pi}{2}}\dfrac{dx}{1+5\cos^2x}$Preview
  43. Q43If $f(x)=\displaystyle\int_{0}^{x}t\cos t\,dt$, then $\dfrac{df}{dx}=$ (a) $x\cos x$ (b) $\cos x-x\sin x$ (c) $x\sin x$ (d) $\sin x+x\cos x$Preview
  44. Q44If $f(x)=\displaystyle\int_{1}^{x}\dfrac{e^{\sin u}}{u}\,du$, $x>1$ and $\displaystyle\int_{1}^{3}\dfrac{e^{\sin x^2}}{x}\,dx=\dfrac12\left[…Preview
  45. Q45Evaluate : $\displaystyle\int_{0}^{\frac{\pi}{2}}\sin^{10}x\,dx$Preview
  46. Q46If $\displaystyle\int_{0}^{\infty}e^{-x}x^n\,dx=5!$, then find the value of $\displaystyle\int_{0}^{\infty}e^{-x}x^{n-1}\,dx$Preview