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

• 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

$$Evaluate\;\int_0^x {\cos t} dt =$$

• A) $$\sin t$$
• B) $${\cos x}$$
• C) $$\sin x$$
• D) $${\cos t}$$

What is the relationship between the area under a curve and the definite integral?

• A) They are unrelated concepts.
• B) The area is the antiderivative of the integral
• C) Both are same concepts.
• D) The area is the derivative of the integral.

$$If\,f(x)\, = \,x^2 ,\,\,then\,\,\int\limits_0^2 {\pi \,[f(x)} ]^2 \,dx\,\,is\,\, - - - - - - -$$

• A) $$\frac{{32\pi }} {5}$$
• B) $$\frac{{17\pi }} {5}$$
• C) $$\frac{{23\pi }} {5}$$
• D) $$\frac{{7\pi }} {5}$$

Which of the following statements is true about $\int\limits_0^1 {(\sin x + \cos x )dx}$?

• A) $$\int\limits_0^1 {(\sin x + \cos x )dx} = [\cos x ]_0^1 + [\sin x ]_0^1$$
• B) $$\int\limits_0^1 {(\sin x + \cos x )dx} = [\sin x]_0^1 - [\cos x]_0^1$$
• C) $$\int\limits_0^1 {(\sin x + \cos x )dx} = - [\sin x]_0^1 + [\cos x]_0^1$$
• D) None

The volume of a cylinder is the area of a cross section of the cylinder multiplied by the ________ of the cylinder.

• A) Diameter
• B) Radius
• C) Base
• D) Height

If we change the letter for the variable of integration but don’t change the limits, then the values of the definite integral will be …………

• A) Changed
• B) Unchanged
• C)
• D)

The value of $\int\limits_0^1 {\frac{{dx}}{{1 + {x^2}}}} \,\_\_\_\_\_.$

• A) $$\frac{\pi }{2}$$
• B) 0
• C) $$\frac{\pi }{4}$$
• D) $$\infty$$