$${\text{The integral }}\int {{{\left( {{x^3} + 1} \right)}^{10}}\,.3{x^2}\,dx\,} {\text{will be equal to ?}}$$

• A) $$- \frac{{{{\left( {{x^3} + 1} \right)}^{11}}}}{{11}} + c$$
• B) $$\frac{{{{\left( {{x^3} + 1} \right)}^{11}}}}{{11}} + c$$
• C) $$\frac{{{{\left( {{x^3} - 1} \right)}^{11}}}}{{11}} + c$$
• D) $${\text{None of these}}$$

The estimated area under f(x) = x^2 from
x = 1 to x = 3 with right end points for
n = 2 is ________.

• A) None of these.
• B) 5.
• C) 10.
• D) 13.

If x = 1 + 2 +3 + 4 + . . . + 20, then
x = ________.

• A) None of these.
• B) 500.
• C) 200.
• D) 210.

Which of the following is the sum of 3k+1 where k goes from 0 to 2?

• A) 30
• B) 27
• C) 33
• D) 39

What will be the sigma notation for 32+52+72+92?

• A) summation of (2k2+1) where (k varies from 3 to 6)
• B) summation of (2k+1)2 where (k varies from 1 to 4)
• C) summation of (k2) where (k varies from 3 to 9)
• D) summation of (2k-1)2 where (k varies from 1 to 4)

f(x) = (x^2)+1 is a continuous function on (-infinity ,+infinity ) , and ‘f’ has no absolute maximum on (-infinity ,+infinity ) because………

• A) lim (f(x)) = +infinity as ‘x’ tends to ±infinity
• B) lim (f(x)) = -infinity as ‘x’ tends to ±infinity
• C) lim (f(x)) = 0 as ‘x’ tends to ±infinity
• D) lim (f(x)) = 1 as ‘x’ tends to ±infinity

Right end point ,left end point, and midpoint evaluation all converges to same result as number of subintervals tends to +ive infinity.

• A) False
• B) True
• C)
• D)

$${\text{For}}\,{\text{Rolle's}}\,{\text{Theorem,}}\,f\,{\text{is}}\,{\text{continuous}}\,{\text{on}}\,{\text{the}}\,{\text{interval __________}}{\text{.}}$$

• A) $$(a,b)$$
• B) $$(a,b]$$
• C) $$[a,b]$$
• D) $$[a,b)$$

In sigma notation 12+14+16+18+20 can be written as……….

• A) summation of (k^2) where (k varies from 6 to10)
• B) summation of (k) where (k varies from 6 to10)
• C) summation of (2k) where (k varies from 1 to5)
• D) summation of (2k) where (k varies from 6 to10)

$\begin{align} & \text{Let a function }f\text{ is twice differentiable at a stationary point }{{x}_{0}}, \\ & \text{then }f\text{ has relative maximum at }{{x}_{0}}\,\text{if} \\ \end{align}$

• A) $\text{ }{f}''({{x}_{0}})\ge 0$
• B) $\text{ }{f}''({{x}_{0}})<0$
• C) $\text{ }{f}''({{x}_{0}})=0$
• D) $\text{ }{f}''({{x}_{0}})>0$

If ‘k’ goes from zero to ‘n’ in the summation of ‘Sink’ then which of the following will be other expression for this given summation such that its lower limit changes to ‘5’?

• A) Summation ‘Sin(k-5)’ while ‘k’ goes from ‘5’ to ‘n-5’
• B) Summation ‘Sin(k+5)’ while ‘k’ goes from ‘5’ to ‘n-5'
• C) Summation ‘Sin(k-5)’ while ‘k’ goes from ‘5’ to ‘n+5’
• D) Summation ‘Sin(k+5)’ while ‘k’ goes from ‘5’ to ‘n+5’

\[{\text{The integral }}\int {\frac{{6x}}{{{{(3{x^2} + 1)}^2}}}\,dx\,} {\text{will be equal to ?}}\]

• A) \[ - \frac{1}{{3{x^2} + 1}} + c\]
• B) \[\frac{1}{{3{x^2} + 1}} + c\]
• C) \[ - \frac{1}{{{{(3{x^2} + 1)}^3}}} + c\]
• D) \[ - \frac{1}{{6x + 1}} + c\]

How is the integral expression for the volume by cylindrical shells affected by changing the axis of revolution?

• A) It changes the variable of integration.
• B) It affects the shape of the region.
• C) It determines the limits of integration.
• D) It doesn't affect the integral setup.

If f(x)= x and g(x)=2x are integrable functions over the interval [0, a] for all xϵ[0, a], then which of the following expressions is true for f and g?

• A) $$\int\limits_0^a {f(x)dx} \leqslant \int\limits_0^a {g(x)dx}$$
• B) $$\int\limits_0^a {f(x)dx} \geqslant \int\limits_0^a {g(x)dx}$$
• C) $$\int\limits_0^a {f(x)dx} < \int\limits_0^a {g(x)dx}$$
• D) $$\int\limits_0^a {f(x)dx} > \int\limits_0^a {g(x)dx}$$

If the integral of f(x) = x and g(x) = 5 from
x = 2 to x = 3 is 5 / 2 and 5 respectively, then the integral of h(x) = x + 5 from x = 2 to x = 3
is ________.

• A) None of these.
• B) 15 / 2.
• C) 5.
• D) 7.

If the solid is revolved around the x-axis and generates a solid with a circular cross section of radius f(x) at x. Then the area of this cross section is

• A) ${\left[ {f(x)} \right]^2}$
• B) $\pi r{\left[ {f(x)} \right]^3}$
• C) $\pi \left[ {f(x)} \right]$
• D) $\pi {\left[ {f(x)} \right]^2}$

$$The\,\,area\,\,of\,\,the\,\,ellipse\,\,\frac{{x^2 }} {{a^2 }}\,\, + \,\,\frac{{y^2 }} {{b^2 }}\,\, = \,\,1$$

• A) $$\pi ab$$
• B) $$\pi (a + b)$$
• C) $$None\,\,of\,\,these$$
• D) $$\frac{1} {4}\,\pi (a^2 + b^2 )$$

The method of slicing by integration is used for finding ----------

• A) surface
• B) volume
• C) area
• D) length

The value of $\int\limits_1^2 {dx = \_\_\_\_\_\_.}$

• A) 1
• B) 0
• C) 2
• D) 3

We can break up definite integrals across a sum or difference $$\int_a^b {f(x) \pm g(x)} dx =$$ as

• A) $$\int_a^b {f(x)dx \pm \int_a^b {g(x)} } dx$$
• B) $$\int_b^a {f(x)dx \pm \int_a^b {g(x)} } dx$$
• C) $$\int_b^a {f(x)dx \pm \int_b^a {g(x)} } dx$$
• D) None of these