Application of Integrals

Find the area of the region bounded by the curve $x = y^2$ and the lines $x = 1, x = 4$ and the $x$-axis in the first quadrant.

Find the area of the region bounded by $y = \sqrt{4-x^2}$, $x=0$ and $x=2$ in the first quadrant.

The area bounded by the curve $y = e^x$, $x$-axis and the lines $x = 0$ and $x = 1$ is:

The area bounded by the line $y = x$, $x$-axis and ordinates $x = -1$ to $x = 2$ is:

Find the area of the region bounded by the parabola $y^2 = 4x$ and the line $y = x$ in the first quadrant, as shown in the figure.

The area bounded by the curve $y = x|x|$, $x$-axis and the ordinates $x = -1$ and $x = 1$ is:

Find the area of the smaller part of the circle $x^2 + y^2 = a^2$ cut off by the line $x = a/\sqrt{2}$.

A parcel of land is enclosed by the curves $y = x^2 + 2$, $y = x$, $x = 0$ and $x = 3$. A surveyor needs to find the area of this plot. What is the calculated area?

An architect is designing a curved roof defined by $y = \sqrt{a^2 - x^2}$ from $x=0$ to $x=a$. If the area under this curve is $25\pi/4$, determine the value of $a$.

A cooling fin for an electronic component is bounded by the parabola $y^2 = 4x$ and its latus rectum. Calculate the area of one face of this fin.

Find the area of the region bounded by the curve $y^2 = x$ and the line $y = \frac{1}{2} x$.

Determine the area bounded by the curve $y = 5x^4$ and the $x$-axis from $x = 0$ to $x = 1$.

The area of the region bounded by $y = \sec^2 x$, the $x$-axis and the lines $x = 0, x = \frac{\pi}{4}$ is:

What is the area of the region bounded by $y = x^2$ and $y = |x|$?

Find the area of the region bounded by the parabola $y^2 = x$ and the line $y = x - 2$.

Determine the area of the region bounded by $y = \sin x$ and $y = \cos x$ for $0 \le x \le 2\pi$.

Find the area of the region bounded by the curves $y = 3x^2$ and $y = 4 - x^2$.

Calculate the area of the region bounded by the curves $y = \sin^{-1}x$, $y = \cos^{-1}x$, and the $x$-axis.

Find the area of the region bounded by the curve $y = 4 - x^2$ and the lines $y = 0$ and $y = 3$.

Find the area of the region bounded by the ellipse $\frac{x^2}{16} + \frac{y^2}{9} = 1$ and the line $x = 2$.