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Mathematics and Statistics · Class 12 Commerce

Ch 7Applications of Definite Integration — Class 12 Mathematics and Statistics, concept-first.

In the previous chapter, a definite integral was evaluated as a single number using the Fundamental Theorem of Integral Calculus, . This Std XII Commerce Mathematics and Statistics chapter gives that number its most important geometric meaning: it measures area.

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Q&A

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Concepts

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Key concepts

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Area under a Curve

The area under a curve , bounded by the X-axis and two vertical lines and , is found by slicing the region into a very large number of extremely thin vertical strips, each of width and height , and adding up their areas.…

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Chapter contents

The NCERT structure, section by section. Open a section to see its questions, then read the concept-first solution.

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Area Under a Curve

In the previous chapter, a definite integral was evaluated as a single number using the Fundamental Theorem of Integral Calculus, .

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Area with Respect to the y-axis

Sometimes a region is described more naturally in terms of the y-axis — bounded by a curve, the y-axis, and two horizontal lines and .

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Regions Below the Axis and the Absolute-Value Rule

The formula was derived assuming the curve lies ABOVE the x-axis. When a curve dips below the x-axis, is negative on that stretch, so the strips have a negative signed height and the definite integral…

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Area Between Two Curves

Definite integration also measures the area of a region trapped between two curves, rather than between one curve and an axis.

Exercises

+Show 6 questions6 questions
  1. Q7Find the area bounded by the curve $y=4x^{3}$, the x-axis, and the ordinates $x=1$ and $x=2$.Free
  2. Q8Find the area bounded by the curve $y=e^{x}$, the x-axis, and the ordinates $x=0$ and $x=1$.Free
  3. Q9Find the area of the region bounded by the parabola $y=9-x^{2}$ and the x-axis.Preview
  4. Q10Find the area of the region bounded by the line $x=2y$, the y-axis, and the horizontal lines $y=1$ and $y=4$.Preview
  5. Q11Find the area of the region enclosed between the parabola $y=x^{2}$ and the line $y=4$.Preview
  6. Q12Explain why the area of a region lying below the x-axis is found by taking the absolute value of the definite integral, and illustrate with…Preview

Sample & Board Papers

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

+Show 7 questions7 questions
  1. Q1The area of the region bounded by the curve $y = x^2$, $x = 0$, $x = 3$, and the X-axis is ______. (a) 9 sq.units (b) $\dfrac{26}{3}$ sq.uni…Preview
  2. Q2Find the area between the two curves (parabolas) $y^2 = 7x$ and $x^2 = 7y$.Preview
  3. Q3Find the area of the region bounded by the parabola $y^2 = 4x$ and the line $x = 3$.Preview
  4. Q4Find the area of the regions bounded by the line $y = -2x$, the X-axis and the lines $x = -1$ and $x = 2$.Preview
  5. Q5The area of the region bounded by the line $y = 4$ and the curve $y = x^2$ is ______. (a) $\frac{32}{3}$ square units (b) 0 square unit (c)…Preview
  6. Q6Find the area of the region bounded by the parabola $y^2 = 25x$ and the line $x = 5$.Preview
  7. Q7Find the area of the region bounded by the curve $x^2 = 16y$ and the line $y = 4$.Preview

More questions

+Show 6 questions6 questions
  1. Example 1Find the area of the region bounded by the curve $y=3x^{2}$, the x-axis, and the ordinates $x=1$ and $x=3$.Free
  2. Example 2Find the area bounded by the line $y=2x+3$, the x-axis, and the ordinates $x=0$ and $x=2$.Free
  3. Example 3Find the area of the region bounded by the parabola $y=x^{2}$, the x-axis, and the ordinates $x=0$ and $x=3$.Preview
  4. Example 4Find the area of the region bounded by the curve $x=y^{2}$, the y-axis, and the horizontal lines $y=0$ and $y=3$.Preview
  5. Example 5Find the area of the region bounded by the curve $y=x^{2}-4$, the x-axis, and the ordinates $x=0$ and $x=2$.Preview
  6. Example 6Find the area of the region enclosed between the line $y=x$ and the parabola $y=x^{2}$.Preview