What is the purpose of using the washer method instead of the disk method?

• A) To reduce the number of slices
• B) To handle shapes with holes.
• C) To simplify the integration process
• D) To increase accuracy

If f(x) and g(x) are constant functions on [a, b], what can be said about the area between the curves?

• A) The area is equal to the absolute difference between the values of f(x) and g(x) over the interval [a, b].
• B) The area is zero.
• C) The area is dependent on the width of the interval [a, b].
• D) The area is infinite.

If the integral of f(x) = x + 1 from
x = 2 to x = 3 is 7 / 2, then the integral of
f(x) = x + 1 from x = 3 to x = 2 is ________.

• A) 7 / 2.
• B) -7 / 2
• C) None of these.
• D) 5 / 2.

What will be the value of $\int\limits_0^2 {(\sin x + 3)dx}$ if $\int\limits_0^2 {(10\sin x + 30)dx = 74}$ ?

• A) 4.2
• B) 5.4
• C) 3.6
• D) 7.4

The volume by the washer perpendicular to the x-axis is

• A) $$\int\limits_a^b {\pi ([f(x)} {]^2} + {[g(x)]^2})dx$$
• B) $\int\limits_a^b {\pi ([f(x)} {]^2} - {[g(x)]^2})dx$
• C) $\int\limits_a^b {\pi ([f(x)} ] + [g(x)])dx$
• D) $\int\limits_a^b {([f(x)} {]^2} - {[g(x)]^2})dy$

$$\begin{gathered} Find\,\,the\,\,area\,\,of\,\,the\,\,region\,\,to\,\,the\,\,left\,\,of\,the\,\,parabola\,\,x = 2y^2 ,\,\,to\,\,the\,\,right\,\,of\,\, \hfill \ the\,y - axis\,\,and\,\,between\,\,y = 1\,\,and\,\,y = 3 \hfill \\ \end{gathered}$$

• A) $$\frac{{52}} {3}$$
• B) $$None\,\,of\,\,these$$
• C) $$\frac{1} {3}$$
• D) $$\frac{{10}} {4}$$

In integration of $f(x)=x{{({{x}^{2}}-3)}^{4}}$ from x=0 to x=2 by substitution method, we take $u={{x}^{2}}-3$ then $du$= ................

• A) $$2xdx$$
• B) $$dx$$
• C) $$2x$$
• D) $$x$$

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• A) The area is decreased.
• B) The area is undefined.
• C) The area is increased.
• D) It doesn't affect the area.

Let f(x)= x and g(x)=2x are integrable functions over the interval [-a, 0] for all xϵ[-a, 0] then which of the following expressions is true for f and g?

• A) $$\int\limits_{ - a}^0 {f(x)dx} < \int\limits_{ - a}^0 {g(x)dx}$$
• B) $$\int\limits_{ - a}^0 {f(x)dx} > \int\limits_{ - a}^0 {g(x)dx}$$
• C) $$\int\limits_{ - a}^0 {f(x)dx} \leqslant \int\limits_{ - a}^0 {g(x)dx}$$
• D) $$\int\limits_{ - a}^0 {f(x)dx} \geqslant \int\limits_{ - a}^0 {g(x)dx}$$

If the upper limit of Definite Integral is equal to its lower limit,then the value of Definite Integral will be _______.

• A) 1
• B) Zero
• C) Same
• D) None of the above

Value of definite integral of f(x)=|2x| from x=-1 to x=0 is ---------

• A) 2
• B) 0
• C) 1
• D) -1

If a solid has a uniform circular cross-section, which method is most appropriate for finding its volume?

• A) Washer Method
• B) Disk Method
• C) Slicing Method
• D) Shell Method

What will be the value of $\int\limits_0^1 {{e^x}dx}$ ?

• A) ec+1
• B) e
• C) ex
• D) 1

If the integral of f(x) = x from x = 1 to x = 3 is 4, then the integral of f(x) = 10x from
x = 1 to x = 3 is ________.

• A) 4.
• B) None of these.
• C) 20.
• D) 40.

The value of $\int\limits_1^{10} {3{x^2}\,dx} \,\_\_\_\_\_\_.$

• A) 999
• B) 333
• C) 33
• D) 99

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

• A) 4
• B) 0
• C) 3
• D) 1

If f(x) is a piecewise function with different expressions on different subintervals, how would you calculate the total area between the curves?

• A) Take the average of f(x) over the interval [a, b].
• B) Use the maximum value of f(x) within the interval.
• C) Ignore the piecewise nature and integrate as a continuous function.
• D) Apply the definite integral separately on each subinterval and sum the results.

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• A) 8
• B) 12
• C) 10
• D) 14

What is the impact of changing the integration limits on the volume determined using the cylindrical shell method.

• A) It may increase or decrease the volume depending on the chosen limits.
• B) It increases the volume.
• C) It decreases the volume.
• D) It has no effect on the volume.

$$If\,the\,curve\,\,over\,\,[a,\,b]\,is\,\,revolved\,about\,y - axis,\,then\,the\,volume\,is\,calculated\,by\,the\,formula\,\, - - - - - - -$$

• A) $$\int\limits_a^b {\pi \,[f(y)} ]^2 \,dy$$
• B) $$\int\limits_a^b {\pi \,[f(x)} ]^2 \,dx$$
• C)
• D)