List of top Mathematics Questions on applications of integrals

Let $f(x) = \frac{1}{20}(x - 5)^2, x \in \mathbb{R}$. If $\int_{-5}^{5} f(x) dx = \int_5^{a} f(x) dx$, where $a > 5$ is a real constant, then the value of $a$ is equal to

Find the area of the region bounded by the curve \(y^2 = 4x\) and the line \(x = 3\).

Find the area of the region bounded by the curve \[ y=x^2-4, \] the \(x\)-axis and the lines \(x=-2\) and \(x=3\).

The area of the region bounded by the \(x\)-axis, the line \(x=4\) and the curve \[ f(x)= \begin{cases} x^2, & 0\le x\le 1\\ \sqrt{x}, & x\gt 1 \end{cases} \] is

A circle passes through the ends of the latus rectum of parabola \( y^2=12x \) and has its centre at the vertex. The area inside the circle and outside the parabola in the \( 1^{st} \) quadrant is:

The area (in sq. units) of the region bounded between the curve \( y=x^{2}+5x+1 \) and the line \( 7x-y+1=0 \) is

The area of the region bounded by the curve $y=x^{2}+x$, the lines $y=x$, $x=1$ and $y=2$ is

The area (in square units) of the region bounded by the parabola $y^2 = 4x$ and the line $y = 2x$ is:

The ratio of areas bounded by the curves \[ y=\cos x \] and \[ y=\cos 2x \] between \[ x=0,\qquad x=\frac{\pi}{3} \] and the \(x\)-axis is:

If the area enclosed between the circle \(x^2+y^2=25\) and the parabola \(y^2=16x\) is \(A\), then \(A=\ ?\)

The area bounded by the X-axis and the curve \[ y=x(x-2)(x+1) \] is:

The line \(y = mx\) bisects the area enclosed by the lines \(x = 0\), \(y = 0\), \(x = \frac{3}{2}\) and the curve \(y = 1 + 4x - x^2\). Then, the value of \(m\) is:

Find the area bounded by the curve \(y=|2-x|\), the \(x\)-axis, and the lines \(x=0\) and \(x=5\).

The whole area surrounded by the curve with the equations $x = a \cos^3 t, y = b \sin^3 t$ is:

The area of the region bounded by the curves \( x = y^2 - 2 \) and \( x = y \) is

$A_1$ is the area bounded by $y=x^2+2$, $x+y=8$, and the $y$-axis in the first quadrant, and $A_2$ is the area bounded by $y=x^2+2$, $y^2=x$, $x=0$ and $x=2$ in the first quadrant. Find $(A_1-A_2)$.

Let \[ f(x)=\int \frac{1-\sin(\ell n t)}{1-\cos(\ell n t)} \, dt \] and \[ f\left(e^{\pi/2}\right)=-e^{\pi/2} \] then find $f\left(e^{\pi/4}\right)$.

The shaded region in the following figure represents a solution set of

Area of the region bounded by the function $f(x)=\begin{cases} x, & x\leq 3 \\ -x+6, & x>3 \end{cases}$ with the $x$-axis (in square units) in the first quadrant is:

The area of the region bounded by \( y = x^{5/2} \) and \( y = x \) (in square units) is

The area of the region bounded by the ellipse \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \) is

The area of the region enclosed by the lines \( 2x + y = 10 \), \( y = 1 \), \( y = 5 \) and the y-axis is

$\displaystyle\lim _{x \rightarrow \infty} \frac{(\sqrt{3 x+1}+\sqrt{3 x-1})^6+(\sqrt{3 x+1}-\sqrt{3 x-1})^6}{\left(x+\sqrt{x^2-1}\right)^6+\left(x-\sqrt{x^2-1}\right)^6} x^3$

The area (in sq. units) of the region bounded by the parabola $x = \frac{y^2}{2}$ and the line $x = y + 4$ is

The area bounded by the curves \( y = \tan x,\ -\frac{\pi}{3} \le x \le \frac{\pi}{3} \), \( y = \cot x,\ \frac{\pi}{6} \le x \le \frac{\pi}{2} \) and the X-axis is: