{\text{Definite integral indicating the arc length of the curve }}y = x^2 {\text{ between }}x = 0{\text{ and }}x = 2{\text{ is }}.........

• A) L = \int\limits_0^2 {\sqrt {1 + 2x^2 } dx}
• B) L = \int\limits_0^2 {xdx}
• C) None of these.
• D) L = \int\limits_0^2 {\sqrt {1 + 4x^2 } dx}

\[ Area\,\,of\,\,the\,\,\,region\,\,bounded\,\,by\,\,the\,\,curves\,\,y = x^2 + 2\,\,,\,\,y = \,\, - x\,\,\,\,;\,\,x = 0\,\,and\,\,x = 1\,\,is \]

• A) \[ \frac{5} {{16}} \]
• B) \[ \frac{17} {{6}} \]
• C) \[ \frac{3} {{16}} \]
• D) \[ None\,\,of\,\,these \]

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

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

{\text{The mean value theorem of }}f(x){\text{ at }}\left[ {x_k ,x_{k - 1} } \right]\,{\text{is }}........

• A) \frac{{f(x_k ) - f(x_{k - 1} )}} {{x_k - x_{k - 1} }} = f'(x^* _k )
• B) \frac{{f({x_k}) + f({x_{k - 1}})}}{{{x_k} + {x_{k - 1}}}} = f'({x_k})
• C)
• D)

\[ If\,f(x)\, = \,\sqrt x ,\,\,then\,\,\int\limits_1^2 {\pi \,[f(x)} ]^2 \,dx\,\,is\,\, - - - - - - - \]

• A) \[ \frac{7 \pi } {2} \]
• B) \[ \frac{3 \pi } {2} \]
• C) \[ \frac{5 \pi } {2} \]
• D) \[ \frac{\pi } {2} \]

When finding the surface area of a solid, what is the purpose of taking the limit of largest width on the x-axis approaches to 0?

• A) To simplify the calculations.
• B) It is not necessary for the surface area calculation.
• C) To obtain a more accurate approximation.
• D) To ensure convergence.

Arc length of the curve y = {x^{3/2}} on [1,3] is _____.

• A) \int\limits_1^3 {\sqrt {1 + {{[\frac{d}{{dx}}\left( {{x^{3/2}}} \right)]}^2}} } dx
• B) \int\limits_1^3 {\sqrt {1 + [\frac{d}{{dx}}\left( {{x^{3/2}}} \right)} ]} dx
• C)
• D)

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

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

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

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

\[ Find\,\,the\,\,area\,\,between\,\,y = x\,\,and\,\,y = \,\, - x(x - 4) \]

• A) \[ None\,\,of\,\,these \]
• B) \[ \frac{9} {2}\, \]
• C) \[ 0 \]
• D) \[ \frac{7} {2}\, \]

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

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

\[ Find\,\,the\,\,area\,\,of\,\,the\,\,region\,\,between\,\,the\,\,x - axis\,,\,\,the\,\,f(x) = \,x^3 - x^2 - 2x;\,\,\, - 1 \leqslant x \leqslant 2 \]

• A) \[ None\,\,of\,\,these \]
• B) \[ \frac{{45}} {4}\, \]
• C) \[ \,\frac{3} {2} \]
• D) \[ \frac{{37}} {{12}} \]

\[ {\text{Length of the curve y = sin(x) from x = 0 to x = }}\pi {\text{ is }}......... \]

• A) \[ \int\limits_0^\pi {\sqrt {1 + \cos ^2 x} } dx \]
• B) \[ \int\limits_0^\pi {\sqrt {1 + \cos x} } dx \]
• C) None of these.
• D) \[ \int\limits_0^\pi {\sqrt {\cos x} } dx \]

\begin{gathered} If\,the\,curve\,\,y\, = \,f(x)\,\,over\,\,[a,\,b]\,is\,\,revolved\,about\,x - axis,\,then\,the\,volume\,is\,calculated\,by\,the\,formula\,\, - - - - - - - \ \end{gathered}

• A) \int\limits_a^b {\pi \,[f(x)} ]^3 \,dx
• B) \int\limits_a^b {\pi \,[f(x)} ]^4 \,dx
• C) \int\limits_a^b {\pi \,[f(x)} ] \,dx
• D) \int\limits_a^b {\pi \,[f(x)} ]^2 \,dx

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

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

\[ {\text{The mean value theorem of }}f(x){\text{ at }}\left[ {x_k ,x_{k - 1} } \right]\,{\text{is }}........ \]

• A) \[ \frac{{f(x_k ) - f(x_{k - 1} )}} {{x_k - x_{k - 1} }} = f'(x^* _k ) \]
• B) $\frac{{f({x_k}) + f({x_{k - 1}})}}{{{x_k} + {x_{k - 1}}}} = f'({x_k})$
• C)
• D)

Area\,\,of\,\,the\,\,\,region\,\,bounded\,\,by\,\,the\,\,curves\,\,y = x^2 + 2\,\,,\,\,y = \,\, - x\,\,\,\,;\,\,x = 0\,\,and\,\,x = 1\,\,is

• A) \frac{5} {{16}}
• B) \frac{3} {{16}}
• C) None\,\,of\,\,these
• D) \frac{17} {{6}}

\[ The\,\,area\,\,of\,\,the\,\,ellipse\,\,\frac{{x^2 }} {{a^2 }}\,\, + \,\,\frac{{y^2 }} {{b^2 }}\,\, = \,\,1 \]

• A) \[ \frac{1} {4}\,\pi (a^2 + b^2 ) \]
• B) \[ None\,\,of\,\,these \]
• C) \[ \pi ab \]
• D) \[ \pi (a + b) \]

\[ Find\,\,the\,\,area\,\,between\,\,y = x\,\,and\,\,y = \,\, - x(x - 4). \]

• A) \[ 0 \]
• B) \[ \frac{9} {2}\, \]
• C) \[ None\,\,of\,\,these \]
• D) \[ \frac{7} {2}\, \]

Use cylindrical shells to find the volume of the solid generated when the region ‘R’ enclosed between \[y = 2x + 1\] and \[y = - 2x - 3\] in the interval [1,3] is revolved about the y-axis is ______.

• A) \[V = \int\limits_1^3 {2\pi x\left( {(2x + 1) + ( - 2x - 3)} \right)} dx\]
• B) \[V = \int\limits_1^3 {2\pi x\left( {(2x + 1) - ( - 2x - 3)} \right)} dx\]
• C)
• D)