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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)

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

• A) $$1 + \cos t$$
• B) $$1 + \cos x$$
• C) $$1 + \cos t$$
• D) $$1 - \cos x$$

What is a cylindrical shell?

• A) A solid by two concentric cylinders.
• B) A sphere with a hole.
• C) A flat, two-dimensional shape.
• D) A solid with a hole in it.

The integral of the sum of two functions is equal to the sum of their integrals.

• A) False
• B) True
• C)
• D)

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) -1+a
• B) -1-a
• C) 1+a
• D) -2

The integral of f(x) =cos(2x) from x=0 to x=pi is ................

• A) 0
• B) None of these.
• C) 2(pi)
• D) 3(pi)

The volume of cylindrical shell for R = 5, r = 3 and h = 2 is -----------

• A) 25*pi
• B) 36*pi
• C) 30*pi
• D) 32*pi

The volume of the sphere with radius r can be calculated by ---------------

• A) 4/3 p r^4
• B) 4/3 p r^2
• C) 4/3 p r
• D) 4/3 p r^3

Mathematically second fundamental theorem of calculus can be written as,

• A) none of these
• B) $$\frac{d} {{dx}}\int_a^t {f(t)dt} = f(t)$$
• C) $$\frac{d} {{dx}}\int_a^t {f(t)dt} = f^/ (x)$$
• D) $$\frac{d} {{dx}}\int_a^x {f(t)dt} = f(x)$$

What will be the average value of y = cos3x with respect to x over [0, 2], if $\int\limits_0^2 {{{\cos }^3}xdx}$ is equal to 0.66?

• A) 1.05
• B) 1.32
• C) 0.33
• D) 1.5

The volume of the solid generated by the region enclosed between $y = \sqrt x$ x=1 , x=3 and x-axis is resvolved by y-axis.Which of the following equation gives the volume of solid by cylindrical shell ____.

• A) $V = \int\limits_1^x {2\pi x\sqrt x dx}$
• B) Both b and c
• C) $V = \int\limits_1^3 {2\pi x\sqrt x dx}$
• D) $V = \int\limits_1^3 {2\pi {x^{\frac{3}{2}}}dx}$

How is the volume of a solid calculated using the method of slicing perpendicular to the y-axis?

• 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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.

When using the technique of slicing to find the volume of an irregular solid, what is the basic idea?

• A) Divide the solid into cubes and sum their volumes.
• B) Divide the solid into cylinders and sum their volumes.
• C) Measure the distance between slices.
• D) Divide the solid into slices and sum their volumes.

$$If\;f\;continuous\;on\;[a,b]\;and\;F(x) = \int_a^x {f(t)dt} ,\;then\;$$

• A) $$F^/ (t) = f(x)\;on\;\;[a,b]$$
• B) $$F^/ (x) = f(x)\;on\;\;[a,b]$$
• C) none of these
• D) $$F^/ (x) = f(t)\;on\;\;[a,b]$$