Q.Find the area bounded by the curve y=sinx, x=0, x=π and the x-axis.

The definite integral $\int_a^b f(x)\,dx$, for $f(x)\geq 0$ on $[a,b]$, equals the exact area of the region bounded by the curve $y=f(x)$, the x-axis, and the ordinates $x=a$ and $x=b$. This follows from the Riemann-sum picture: split $[a,b]$ into $n$ thin strips of width $\Delta x=(b-a)/n$, erect a rectangle of height $f(x_i)$ on each strip, and sum their areas $\sum f(x_i)\Delta x$. As $n\to\infty$, the flat-topped rectangles hug the curve more and more closely and the sum's limit is, by definition, the integral -- so the …