EVERY continuous function on an interval has an anti-derivative ……………

• A) outside of interval
• B) On that interval
• C)
• D)

F(x) (antiderivative of f) was determined to be 4x+C where C denotes the .............

• A) Differentiation constant.
• B) Integration constant.
• C)
• D)

Which of the following statements is true about $\int\limits_0^1 {(\sin x - {{\sec }^2}x)dx}$?

• A) $$\int\limits_0^1 {(\sin x - {{\sec }^2}x)dx} = [\cos x]_0^1 - [\tan x]_0^1$$
• B) $$\int\limits_0^1 {(\sin x - {{\sec }^2}x)dx} = [\cos x]_0^1 + [\tan x]_0^1$$
• C) None
• D) $$\int\limits_0^1 {(\sin x - {{\sec }^2}x)dx} = - \,[\cos x]_0^1 - [\tan x]_0^1$$

Why can't we use the formula for the volume of a right cylinder to find the volume of irregular 3D solids?

• A) Irregular solids have infinite height.
• B) Irregular solids are not made up of finitely many right cylinders.
• C) Irregular solids have cylinderical surfaces.
• D) The formula for right cylinders is not accurate.

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• A) data:image/png;base64,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.
• B) data:image/png;base64,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.
• C) data:image/png;base64,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.
• D) data:image/png;base64,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.

$$The\,\,region\,\,bounded\,\,by\,\,the\,\,curves\,\,y = \sqrt x \,\,,\,\,y = 1\,\,and\,\,x = 4$$

• A) $$\frac{1} {2}$$
• B) $$None\,\,of\,\,these$$
• C) $$0\,\,$$
• D) $$\,\,\frac{5} {3}$$

The volume of cylindrical shell for R = 3, r = 2 and h = 1 is -----------

• A) 6*pi
• B) 4*pi
• C) 5*pi
• D) 7*pi

The Volume of a cylindrical shell is _____.

• A) None of the above
• B) $2\pi .(\frac{{{r_2} + {r_1}}}{2}).h.({r_2} + {r_1})$
• C) $2\pi .(\frac{{{r_2} + {r_1}}}{2}).h.({r_2} - {r_1})$
• D) $\pi .(\frac{{{r_2} + {r_1}}}{2}).h.({r_2} - {r_1})$

For the adjacent intervals, [a,c] and [c,b],where c is any number,$$\int_a^b {f(x)} dx =$$

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

If integral of ‘f(x)’ from [1,2] = 5 ,and integral of ‘f(x)’ from [3,2] = 4 , than integral of ‘f(x)’ from [1,3] is ……….

• A) 3
• B) 9
• C) 1
• D) 2

The integral of f(x) =sin(x+1) from x=0 to x=1 is ..........

• A) sin(1)-sin(2)
• B) cos(2)-1
• C) cos(1)-cos(2)
• D) cos(2)-cos(1)

If integral of ‘f(x)’ from [-8,16] = 54 , then integral of ‘5f{x)’ from [-8,16] =………..

• A) 240
• B) 270
• C) 220
• D) 260

The volume of a cylindrical shell with an outer cylinder radius R=3, inner cylinder radius r=2, and length h=2 is……..

• A) data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAC8AAAAWCAYAAABQUsXJAAAAAXNSR0IArs4c6QAAAARnQU1BAACxjwv8YQUAAAAJcEhZcwAADsMAAA7DAcdvqGQAAAElSURBVFhH7ZQBDYMwEEVnAAlIQAIWcIAFHKAABTjAAAYwgAEcIOKW13BJ15UuIR3dkr6ELLRNeNf7t4f8MVk+FVk+FVk+FUH5YRik7/vj7ZV5nqWqKimKwvxO03Ts3IdXfl1XqevaiFGAy7IsZk+FOcM763fyJr9tm7RtK/u+S9M0Xvmu60xxNmVZmvU7CcbmTJ6YuHHiLI/CPt04e2J06ZI8H3fXOcs6jONo5JgLugh0kn06G4uo8kTHhtunEGA+6FhMLskj+Sk2wFxoPJiH2DNxSZ4ouAPrzgHx0BjBN/5Og/II+m6LLCOmMkSDbth5pmi7E5ynCxRI/mPgleej3BQf5KEINybIIaz79r8HRbCneQfOsBbz9oM3/+tk+VRk+TSIPAEyIkhPU2vczwAAAABJRU5ErkJggg==.
• B) data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAABwAAAAVCAYAAABVAo5cAAAAAXNSR0IArs4c6QAAAARnQU1BAACxjwv8YQUAAAAJcEhZcwAADsMAAA7DAcdvqGQAAACpSURBVEhL7ZJRDQMhEERrAAknAQlYwAEWcIACFOAAAxhABw4Qsc1srvzc8dGwbZqGl5AchPAys/egL7OF4myhOL8hDCGQUmq6aq3nzfe5CFNK/GAphZxzfNZ7Z1FrjfcrTCtFSshBzpm01vy9ylRojBnVee95SXArRHWo8AXSIaUEt8IYI1lrzx2NHwU1Y54rXIRIdxzHmB9AvTiTSDmd4afYQnH+XUj0BNGx4soTqKjgAAAAAElFTkSuQmCC.
• C) data:image/png;base64,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.
• D) data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAACoAAAAbCAYAAAAK5R1TAAAAAXNSR0IArs4c6QAAAARnQU1BAACxjwv8YQUAAAAJcEhZcwAADsMAAA7DAcdvqGQAAAD5SURBVFhH7ZNdDYQwEAbPABKQgAQs4AALOEABCnCAAQxgAAM4QMRepte99H564WE5AukkhHbhYb529yYnIYlak0StSaLWXFO0aRq/+j9R0a7rJMuyj+cofopWVeV3x3N90XmepSiKr+3BYx0yKjoMg+R5/pThPY6j+7auq7Rt695lWTppoLbXwEVFQxBCAOFlWXz1USeMQhgC7sEmUUAK0b7vfUXcCesVE+A9iCWbRVUkFOWU6WXgJDnRvYiKIhX2JFJcM2utsZ+mye11+OjXMIwVUdG6rl+mmr0ODTA4YX8Siv+Q1TCWbL76o0mi1iRRa5KoNScRFbkDKiUX6vIFfNsAAAAASUVORK5CYII=.

Let R be the plane region bounded above by a continuous curve y=f(x) below by the x-axis and on the left and right, respectively, by the lines x=a and x=b the volume of the solid generated by revolving R about the y-axis is given by ______.

• A) $$V = \int\limits_a^b {2\pi xf(x)dx}$$
• B) None of the above
• C) $$V = \int\limits_a^b {2\pi f(x)dx}$$
• D) $$V = \int\limits_a^b {2\pi xdx}$$

If “f” is a continuous function on [a,b] then $$\int\limits_a^b {f(x)\,dx = } \,\_\_\_\_\_\_.$$

• A) $$\int\limits_{b}^{b}{f(x)dx}$$
• B) $$- \int\limits_c^a {f(x)\,dx}$$
• C) $$- \int\limits_b^a {f(x)\,dx}$$
• D) $$- \int\limits_a^b {f(x)\,dx}$$

$\int_a^b {f(x)} dx = 0$ if__________.

• A) None of these
• B) $a = b$
• C) $a < b$
• D) $a > b$

The volume of the solid generated by the region enclosed between $y = \sqrt x$ x=1 , x=4 and x-axis is revolved about y-axis.Which of the following equation gives the volume of solid by cylindrical shell ____.

• A) 77.91
• B) $$\frac{{124\pi }}{5}$$
• C) $$\frac{{114\pi }}{5}$$
• D) Both a and c

Definite integral from "a" to "b" of f(x)is equal to the ..........of f(x)evaluated at the "b" and then at "a".

• A) Indefinite Integral
• B) Derivative
• C) None of these
• D) Both of these.

If f(x)= x and g(x)=x+1are integrable functions over the interval [a, b] for all xϵ[a, b], then which of the following expressions is true for f and g?

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

$$Area\,\,between\,\,the\,\,x - axis\,\,and\,\,the\,\,curve\,\,y = \,\cos x;\,\,when\,\,0 \leqslant x \leqslant 2\pi \,\,is\,$$

• A) $$None\,\,of\,\,these$$
• B) $$\,2$$
• C) $$\,0\,$$
• D) $$\,4$$