Distance formula is based on the............

• A) Pythagoras theorem
• B) Mean value theorem
• C) None of these.
• D) Intermediate value theorem.

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

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

The\,\,area\,\,bounded\,\,by\,\,the\,\,parabola\,\,y^2 \,\, = \,\,x\,,\,\,st.line\,\,y = 4\,\,and\,\,y - axis\,\,is

• A) \frac{{64}} {3}
• B) 7\sqrt 2 \,
• C) \frac{{16}} {3}
• D) None\,\,of\,\,these

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• A) The area is undefined.
• B) The area is zero.
• C) The area is dependent on the width of the interval [a, b].
• D) The area is infinite.

What is the result of rotating a polygonal path around the x-axis?

• A) A circle
• B) A solid of revolution
• C) A cone
• D) A frustum

The graphs of the smooth functions are _____.

• A) Smooth Curves
• B) Straight Lines
• C) Polygon
• D) Smooth Graph

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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 length of the WHOLE polygonal path will be _______.

• A) \sum\limits_{k = 1}^n {{L_k}} = \sum\limits_{k = 1}^n {\sqrt {1 + (f'({x_k})} {)^2}}
• B) \sum\limits_{k = 1}^n {{L_k}} = \sum\limits_{k = 1}^n {\sqrt {1 + (f'({x_k})} {)^2}\Delta {x_k}}
• C) {\text{None}}\,{\text{of}}\,{\text{the}}\,\,{\text{above}}
• D) \sum\limits_{k = 1}^n {{L_k}} = \mathop {\lim }\limits_{\max \,\Delta x \to 0} \sum\limits_{k = 1}^n {\sqrt {1 + (f'({x_k})} {)^2}} dx

The ........ states that if f(x)is continuous on the closed interval [a,b]and differentiable on the open interval (a,b)then there exists a point c such that
[f(b)-f(a)]/(b-a)=f'(c)

• A) Extreme value theorem
• B) Intermediate value theorem
• C) Mean value theorem
• D) None of these

Definite integral indicating the arc length of the curve y=coshx between x=0 and x=a is .........

• A) \int\limits_0^a {\cosh xdx}
• B) None of these.
• C) \int\limits_0^a {1 + \sinh xdx}
• D) \int\limits_0^a {\sinh xdx}

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

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

If the function f(x) is not smooth or differentiable, how would it affect the surface area calculation?

• A) The surface area becomes infinite.
• B) The surface area calculation is not defined.
• C) It has no effect.
• D) The calculation becomes more accurate.

\[ If\,f(x)\, = \,x^2 ,\,\,then\,\,\int\limits_0^2 {\pi \,[f(x)} ]^2 \,dx\,\,is\,\, - - - - - - - \]

• A) \[ \frac{{7\pi }} {5} \]
• B) \[ \frac{{17\pi }} {5} \]
• C) \[ \frac{{23\pi }} {5} \]
• D) \[ \frac{{32\pi }} {5} \]

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

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

What is the formula to calculate the lateral surface area of a frustum cone when the radii of the top and bottom circles are equal?

• A) 2πrl
• B) πrl
• C) 24πrl
• D) 3πrl

Evaluate\;\frac{d} {{dx}}\int_2^x t dt

• A) x
• B) x^3
• C) x^2
• D) None of these

Which of the following statements is true regarding the ARC LENGTH PROBLEM?

• A) It focuses on finding the area under a curve.
• B) It is applicable only to polynomial functions.
• C) It addresses the challenge of determining the length of curved lines.
• D) It only deals with straight lines.

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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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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)

Length of the arc y=c from x=0 to x=1 is _____.

• A) None of the above
• B) 1
• C) -1
• D) 0