List of top Mathematics Questions on Application of Integrals

Which of the following expressions will give the area of the region bounded by the curve $y = x^2$ and the line $y = 16$?

Find the total area of the region bounded between the curve \( y = x^3 \), the x-axis, and the vertical lines \( x = -1 \) and \( x = 1 \).

The area of the region bounded by the curves \(x + 3y^2 = 0\) and \(x + 4y^2 = 1\) is equal to:

If the area of the region bounded by \(16x^2 - 9y^2 = 144\) and \(8x - 3y = 24\) is \(A\), then \(3(A + 6 \log_e(3))\) is equal to _______.

The area of the region \(\{(x, y): y \le \pi - |x|, y \le |x \sin x|, y \ge 0\}\) is:

Find the total area of the region bounded by the parabola \( y^2 = 4x \) and the straight line \( y = x \).

Find the total area of the bounded region enclosed between the parabola curve \( y^2 = 8x \) and its vertical latus rectum boundary line.

Find the area of the region enclosed by the ellipse given parametrically by the coordinates \( x = 2\sin\theta \) and \( y = 3\cos\theta \) where \( 0 \le \theta \le 2\pi \).

Find the area of the region bounded by the curve \( y^2 = x \) and the line \( x = 4 \):

The area of region bounded by the curve \[ y^2=4ax \] and the straight line \[ x=2a,\qquad a>0 \] in the first quadrant is:

The area bounded by \[ 2 - 4x \leq y \leq x^2 + 4 \quad \text{and} \quad x = \frac{1}{2}, \, x \geq 0, \, y \geq 0 \] (in square units) is:

The area of the region, inside the ellipse \(x^2+4y^2=4\) and outside the region bounded by the curves \(y=x-1\) and \(y=1-x\), is:

If the area of the region \( \{ (x, y) : x^2 + 1 \leq y \leq 3 - x \} \) is divided by the line \( x = -1 \) in the ratio \( m : n \) (where \( m \) and \( n \) are coprime natural numbers), then the value of \( m + n \) is:

Area enclosed by \[ x^2 + 4y^2 \leq 4, \quad |x| \leq 1, \quad y \geq 1 - |x| \text{ is} \]

The area (in sq. units) of the region bounded by the parabola y2 = 4x and the line x = 1 is

The area (in sq. units) of the region bounded by the curve \( y = x^5 \), the x-axis and the ordinates x = -1 and x = 1 is equal to

The area (in sq. units) of the region bounded by the parabola y2 = 4x and the line x = 1 is

\[ \int \frac{x + 5}{(x + 6)^2} e^x \, dx \] is equal to:

The area of the region bounded by the curve $y^2 = x$ between $x = 0$ and $x = 1$ is:

Let $f'(x) = 3(x^2 + 2x) - \frac{4}{x^3} + 5$, $f(1) = 0$. Then, $f(x)$ is:

The area (in sq. units) of the region bounded by the curve \( y = x^5 \), the x-axis and the ordinates x = -1 and x = 1 is equal to

Evaluate \[ \int_0^{\frac{\pi}{2}} \frac{x}{\sin x + \cos x} \, dx. \]

Find \[ \int \frac{x^2 + 1}{(x - 1)^2 (x + 3)} \, dx. \]