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,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.
• 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) $$\,4$$

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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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.

Derivative of f(x)= a-7,where a is a constant is....

• A) 0
• B) a-7
• C) -7
• D) a

If sinx=t then dt=.....

• A) sinxdx
• B) -cosxdx
• C) cosxdx
• D) sinxdt

Antiderivative of cosx is ..........

• A) sinx+c
• B) xsinx
• C) cosx+sinx
• D) None of these.

While calculating Riemann sum it is necessary to take equal length subintervals.

• A) False
• B) True
• C)
• D)

$$Area\,\,lying\,\,between\,\,the\,\,parabola\,\,y^2 = \,\,4ax\,\,and\,\,its\,\,latus\,\,rectum\,\,is\,$$

• A) $$\frac{8} {3}a^2$$
• B) $$\,\frac{8} {3}a\,$$
• C) $$None\,\,of\,\,these$$
• D) $$\frac{4} {3}a$$

By using cylindrical shells to find the volume of the solid when the region R in the first quadrant enclosed between $y = x$ and $$y = {x^2}$$ is revolved about the y-axis is ______.

• A) pi/6
• B) pi/3
• C)
• D)