List of top Mathematics Questions on applications of integrals

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

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

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

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

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:

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

\(I(x) = \int \frac{x^2(xsec^2x + tanx)}{(xtanx+1)^2} dx\). If \(I(0)=1\) then \(I(\frac{\pi}{4})\) is equal to

The area enclosed by \(y = |x-1| + |x-2|\) and \(y = 3\)

Find area bounded by the curves \(y\) = \(max\{sin x, cos x\}\) and \(x-axis\) between \(x = -  \pi  \)and \(x =   \pi\)

Let $\alpha$ be the area of the larger region bounded by the curve $y^2=8 x$ and the lines $y=x$ and $x=2$, which lies in the first quadrant Then the value of $3 \alpha$ is equal to _____

Let \(\sum_{n=0}^{\infty }\frac{n^{3}((2n)!+(2n-1)(n!))}{(n!)((2n)!)}=ae+\frac{b}{e}+c\), where $a, b, c \in Z$ and $e=\displaystyle\sum_{n=0}^{\infty} \frac{1}{n !}$ Then $a^2-b+c$ is equal to _______

The area of the region described by \( A = \{(x,y) : x^2 + y^2 \le 1 \text{ and } y^2 \le 1 - x \} \) is

if \(∫ \frac{(x^2+1)e^x}{(x+1)^2}dx = ƒ(x)e^x+C\) where \(C\) is a constant, then \(\frac{d^3ƒ}{dx^3}\) at \(x = 1\) is equal to :

If $P$ is a point on the parabola $y=x^{2}+4$ which is closest to the straight line $y =4 x -1$, then the co-ordinates of $P$ are

If the curve $y=a x^{2}+b x+c, x \in R,$ passes through the point (1,2) and the tangent line to this curve at origin is $y = x ,$ then the possible values of $a , b , c$ are :