What is the formula for finding the volume of a cylindrical shell? where R is the radius of the outer cylinder, r is the radius of the inner cylinder and h is the length of the cylindrical shell.

• A) data:image/png;base64,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.
• B) data:image/png;base64,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.
• C) data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAIIAAAAiCAYAAACTHwETAAAAAXNSR0IArs4c6QAAAARnQU1BAACxjwv8YQUAAAAJcEhZcwAADsMAAA7DAcdvqGQAAAL3SURBVHhe7ZgNkcJADEbPABKQgAQs4AALOEABCnCAAQxgAAM4QETvXobM5DK7dNttC4W8mZ3rbzZNvs2G+2mC4I8QQiCEEAIhhBAIIYRACCEEQgghEEIIgRBCCIQQQiB8tRDu93tzOBya5XLZLBYLOa7hdrs1u91ObGHzfD4/7rw/Xy2Ey+XSbLdbOSZpJLAGhHQ8HuV4v983m81GjufAxwhBk3q9Xh9XusFqHjJxp9PpX4XBN86pQiXwHVqppiApBIKKAzoUXTWM9Xr9uPp6CDr+aJBtEO14JhRKOmIYAvzQSmOhWlg/2xiiSpWSrQiUNpxAFBZWzWq1GixoteAfSffBZfXZIPIcfjM8fGvfSuLBD0SVSzZ+pUSSQhfkFGSFoE74DyKQXhyvBGHqvmzxQoDUNSsCjmvwIkjZ4x4+lMTwrYRgHSaQJcHivbaBrRyU+lRp16HvUpU4T1WnnBCwqzCPteufT8FC4DniQiw41t5CfzHoyPUcVASebUNzgHiwxTHzD1W9LMVCYHKc8BViaJiPFc48NmnsrSTOoolM4YWAXex5G13RuJAY9nASmkt4Dnwr6bF0Lq0yCJ5vqK1cKbJC8CUMx/V4Cgiy7qW5lU9Ac0lQIegggNgsRZOgA3v2ehdbHi/SHDqXXXx8b1fhlZAVAuAEzuB4FxXyXtvQwOZgPn2GVZxq8rjfJgQgkAi5S8eeQ5PD375Y356hc1leJgTK9BRbgoc5NdiUxtSeSkBzJdYHm62N89KOPcfXCqH2o/vgA8CHEzwqg/WlS48Aeq2mT1DfamKCH1bA2EslV/+P8PKtgQnHaEzaIEh25Wp37n3J9Q6sfnoC7vktBdspW6Vgj/ex37d7J662wmHPJzf1DVRnzhm1Ta/nqRDmAAFkhc0FVjeJnLrKtjF7IRBQVs7UPUxfqES1fcoYzF4IQJkc4hfB2Lyznx8hBKAysNL67ttjQ0/AFvauYv0YIQR1hBACIYQQCCGEQAghBEIIIRBCCMEfTfMLZhVGP2cWQMYAAAAASUVORK5CYII=.
• D) data:image/png;base64,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.

We can approximate integrals by Riemann sums.

• A) True
• B) False
• C)
• D)

If the definite integral of f(x)=Sin x over the interval [-a,0] is equal to ‘- 2’ then what will be the value of the definite integral of f(x)= (Sin x) +1 over the same interval?

• A) -1
• B) -2+a
• C) 2-a
• D) -2-a

The integral of f(x) =sin(x) from x=0 to x=pi is ...............

• A) 0
• B) None of these
• C) 2
• D) 1

The value of $\int\limits_1^2 {\ln x\,dx} \,\_\_\_\_.$

• A) 2ln2-1
• B) 2ln2+2
• C) 2ln2+1
• D) ln2-1

If the definite integral of f(x)=cos x over the interval [0, a] is equal to ‘1’ then what will be the value of a?

• A) 2π
• B) π
• C) π/2
• D) π/4

If the function f(x) is negative for some values in the interval [a, b], what alteration should be made to the volume formula?

• A) Take the absolute value of f(x).
• B) Exclude those values from the interval
• C) Multiply by -1
• D) No modification needed.

$\int\limits_a^b {f(x)dx = \_\_\_\_\_\_.}$

• A) $- \int\limits_b^a {f(x)dx}$
• B) $\int\limits_b^a {f(x)dx}$
• C)
• D)

The ........ states that if f(x)is continous on the closed interval [a,b]and differentiable on the open interval (a,b)then there exists a point c such that
[f(b)-f(a)]/(b-a)=f'(c)

• A) Mean value theorem.
• B) None of these.
• C) Intermediate value theorem .
• D) Extreme value theorem.

The volume by the Washers generated by revolving the region around y-axis is given by the formula ----------

• A) $$\int\limits_a^b {\pi \,([u(y)} ]^2 \, * \,[v(y)]^2 )dy$$
• B) $$\int\limits_a^b {\pi \,([u(y)} ]^2 \, - \,[v(y)]^2 )dy$$
• C) $$\int\limits_a^b {\pi \,[\frac{{u(y)}} {{v(y)}}} ]^2 \,dy$$
• D) $$\int\limits_a^b {\pi \,([u(y)} ]^2 \, + \,[v(y)]^2 )dy$$

The volume of solid obtained when the region under the curve y=x over the interval [0,2] is revolved about the x-axis is

• A) $V = \int\limits_0^2 {{x^2}dx}$
• B) $V = \int\limits_0^2 {\pi {y^2}dy}$
• C) $V = \int\limits_0^2 {\pi {x^2}dx}$
• D) $V = \int\limits_0^2 {\pi xdx}$

If the average value of y = sin3x with respect to x over [0, 2] is 0.525, then what will be the value of $\int\limits_0^2 {{{\sin }^3}xdx}$?

• A) 0.5
• B) 2.5
• C) 1.5
• D) 1.05

If the definite integral of f(x)=cos x over the interval [-a,0] is equal to ‘-1’ then what will be the value of the definite integral of f(x)= (cos x)+1 over the same interval?

• A) 0
• B) 1+a
• C) -1+a
• D) -1-a

The volume of the solid generated by the region enclosed between $y = \sqrt x$ x=0 , x=1 and x-axis is revolved about y-axis.Which of the following equation gives the volume of solid by cylindrical shell ____.

• A) $\frac{{4\pi }}{5}$
• B) $\frac{{124\pi }}{5}$
• C) None of the above
• D) $\frac{{12\pi }}{5}$

We cannot evaluate definite integrals by substitution method.

• A) True
• B) False
• C)
• D)

$$Area\,\,between\,\,the\,\,curves\,\,y = 4 + 3x - x^2 \,\,and\,\,x - axis\,\,in\,\,sq.\,\,unit\,\,is\,\,$$

• A) $$\frac{{125}} {6}$$
• B) $$\frac{{125}} {4}$$
• C) $$None\,\,of\,\,these$$
• D) $$\frac{{125}} {3}$$

If the radius of a sphere is doubled, how does the volume change?

• A) It becomes half of the original volume.
• B) It remains the same of the original volume.
• C) It becomes eight times of the original volume.
• D) It becomes four times of the original volume.

$$The\,\,area\,\,bounded\,\,by\,\,the\,\,curve\,\,y = \,4x - x^2 \,\,and\,\,x - axis\,\,is\,\,$$

• A) $$\frac{{30}} {7}\,\,$$
• B) $$None\,\,of\,\,these$$
• C) $$\frac{{31}} {7}\,\,$$
• D) $$\frac{{32}} {3}$$

What is the geometric shape used in the method of cylindrical shells for finding volumes?

• A) Sphere.
• B) Cylinder.
• C) Pyramid.
• D) Cone.

If $$\int_0^2 {(x^2 + 1)} dx = \frac{{14}} {3}$$ Then the solution of $$\int_2^0 {(x^2 + 1)} dx =$$ will be....

• A) 14/3
• B) none of these
• C) -14/3
• D) -3/14