List of top Multivariable Calculus Questions on Integral Calculus

The value of
\(\lim\limits_{t→\infin}\left((\log(t^2+\frac{1}{t^2}))^{-1} \int\limits_{1}^{\pi t} \frac{\sin^25x}{x}dx\right)\)
equals ___________ (rounded off to two decimal places).

The area of the region
\(R=\left\{(x,y) \in \R^2\ : 0\le x \le1,0 \le y \le 1\ \text{and}\ \frac{1}{4} \le xy \le \frac{1}{2} \right\}\)
is __________ (rounded off to two decimal places).

Let T be the planar region enclosed by the square with vertices at the points (0,1), (1,0), (0, −1) and (−1,0). Then, the value of
\(\iint\limits_T\left(\cos(\pi(x-y))-\cos(\pi(x+y))\right)^2dx\ dy\)
equals ____________ (rounded off to two decimal places).

Let
S = {(x, y, z) ∈ \(\R^3\) : x² + y² + z² < 1}.
Then, the value of
\(\frac{1}{\pi}\iiint_s\left((x-2y+z)^2+(2x-y-z)+(x-y+2z)^2\right)dxdydz\)
equals ________ (rounded off to two decimal places).

For n ∈ \(\N\), if
\(a_n=\frac{1}{n^3+1}+\frac{2^2}{n^3+2}+...+\frac{n^2}{n^3+n}\)
then the sequence \(\left\{a_n\right\}_{n=1}^{\infin}\) converges to ____________ (rounded off to two decimal places)

Let c > 0 be such that
\(\int^c_0e^{s^2}ds=3\)
Then, the value of
\(\)\(\int\limits_{0}^c\left(\int\limits^c_0 e^{x^2+y^2}dy\right)dx\)
equals __________(rounded off to one decimal place).

The area of the curved surface
\(S = \left\{ (x, y, z) ∈ \R^3 : z^2 = (x − 1)^2 + (y − 2)^2 \right\}\)
lying between the planes z = 2 and z = 3 is

Suppose f : (−1, 1) → \(\R\) is an infinitely differentiable function such that the series \(\sum\limits_{j=0}^{\infin}a_j\frac{x^j}{j^!}\) converges to f(x) for each x ∈ (−1, 1), where,
\(a_j=\int\limits_{0}^{\pi/2}\theta^j\cos^j(\tan\theta)d\theta+\int\limits^{\pi}_{\pi/2}(\theta-\pi)^2\cos^j(\tan\theta)d\theta\)
for j ≥ 0. Then

Let
\(f(x,y)=\iint\limits_{(u-x^2)+(v-y)^2 \le 1}e^{-\sqrt{(u-x)^2+(v-y)^2}}du\ dv.\)
Then \(\lim\limits_{n \rightarrow \infin}f(n,n^2)\) is

The limit
\(\lim\limits_{a\rightarrow0}\left(\frac{\int\limits_{0}^{a}\sin(x^2)dx}{\int\limits_{0}^{a}(\ln(x+1))^2dx}\right)\)
is

Which of the following functions is/are Riemann integrable on [0, 1] ?

The value of
\(\lim\limits_{n\rightarrow \infin}\left(n\int\limits^1_0\frac{x^n}{x+1}dx\right)\)
is equal to __________ . (rounded off to two decimal places)

Let V be the volume of the region S ⊆ \(\R^3\) defined by
S = {(x, y, z) ∈ \(\R^3\) : xy ≤ z ≤ 4, 0 ≤ x² + y² ≤ 1}.
Then \(\frac{V}{π}\) is equal to ________ . (rounded off to two decimal places)

The sum of the series \(\sum\limits_{n=1}^{\infin}\frac{2n+1}{(n^2+1)(n^2+2n+2)}\) is equal to _________.(rounded off to two decimal places)

Let S be the triangular region whose vertices are (0, 0), \((0,\frac{\pi}{2})\) and \((\frac{\pi}{2},0)\). The value of \(\iint\limits_S\sin(x)\cos(y)dx\ dy\) is equal to ________. (rounded off to two decimal places)

Let H: \(\R\rightarrow\R\) be the function given by H(x) = \(\frac{1}{2}(e^x+e^{-x})\) for \(x\isin\R\).
Let f: \(\R\rightarrow\R\) be defined by
\(f(x)=\displaystyle\int_{0}^{\pi}H(xsin\theta)d\theta\) for \(x\isin\R\)
Then which one of the following is true?

Let a, b be positive real numbers such that \(a\lt b\). Given that
\(\lim\limits_{N\rightarrow\infin}\displaystyle\int^{N}_{0}e^{-t^2}dt=\frac{\sqrt \pi}{2},\) the value of
\(\lim\limits_{N\rightarrow\infin}\displaystyle\int^{N}_{0}\frac{1}{t^2}(e^{-at^2}-e^{-bt^2})dt\) is equal to

For \(-1\le x\le1\), if f (x) is the sum of the convergent power series \(x+\frac{x^2}{2^2}+\frac{x^3}{3^2}+...+\frac{x^n}{n^2}+...\) then \(f(\frac{1}{2})\) is equal to

For \(n\isin\N\) and \(x\isin[1,\infin)\), let
\(f_n(x)=\int\limits_{0}^{\pi}(x^2+(cos\theta)\sqrt{x^2-1})^nd\theta\)
Then which one of the following is true?

Let P be a 3×3 real matrix having eigenvalues \(\lambda_1\) = 0, \(\lambda_2\) = 1 and \(\lambda_3\) = −1. Further, \(v_1=\begin{pmatrix} 1 \\ 0 \\ 0 \end{pmatrix}\), \(v_2=\begin{pmatrix} 1 \\ 1 \\ 0 \end{pmatrix}\) and \(v_3=\begin{pmatrix} 1 \\ 0 \\ 1 \end{pmatrix}\) are eigenvectors of the matrix P corresponding to the eigenvalues \(\lambda_1\), \(\lambda_2\) and \(\lambda_3\), respectively. Then the entry in the first row and the third column of P is

The value of the limit
\(lim _{n→∞}(\frac{1^4+2^4+…+n^4}{n^5}+\frac{1}{√n}(\frac{1}{\sqrt{n+1}}+\frac{1}{\sqrt{n+2}}+...+\frac{1}{\sqrt{4n}}))\)
is equal to___(Rounded off two decimal places)

If G is the region in \(\R^2\) given by
\(G=\left\{(x,y)\in \R^2:x^2+y^2<1,\frac{x}{\sqrt3}< y < \sqrt3x,x>0,y>0\right\}\)
then the value of
\(\frac{200}{\pi}\iint_Gx^2dx\ dy\)
is equal to _________. (Rounded off to two decimal places)

Let f : [0, π] → \(\R\) be the function defined by
\(f(x)=\begin{cases} (x-\pi)e^{\sin x} & \text{if } 0 \le x \le \frac{\pi}{2},\\ xe^{\sin x}+\frac{4}{\pi} & \text{if }\frac{\pi}{2}\lt x \le \pi. \end{cases}\)
Then the value of
\(\int\limits_0^{\pi}f(x)dx\)
is equal to _________. (Rounded off to two decimal places)

Define f : \(\R^2 → \R\) by
f(x, y) = x² + 2y² - x for (x, y) ∈ \(\R^2\).
Let D = {(x, y) ∈ \(\R^2\) ∶ x² + y² ≤ 1} and \(E=\left\{(x,y)\in \R^2:\frac{x^2}{4}+\frac{y^2}{9} \le 1\right\}.\)
Consider the sets
D_max = {(a, b) ∈ D ∶ f has absolute maximum on D at (a, b)}, 
D_min = {(a, b) ∈ D ∶ f has absolute minimum on D at (a, b)}, 
E_max = {(c, d) ∈ E ∶ f has absolute maximum on E at (c, d)}, 
E_min = {(c, d) ∈ E ∶ f has absolute minimum on E at (c, d)}.
Then the total number of elements in the set
D_max ∪ D_min ∪ E_max ∪ E_min
is equal to _________.

If f : [0, ∞) → \(\R\) and g : [0, ∞) → [0, ∞) are continuous functions such that
\(\int^{x^3+x^2}_0f(t)dt=x^2\) and \(\int_0^{g(x)}t^2dt=9(x+1)^3\) for all x ∈ [0, ∞),
then the value of
f(2) + g(2) + 16 f(12)
is equal to _________. (Rounded off to two decimal places)