This set of Differential and Integral Calculus Questions and Answers for Freshers focuses on “Rectification”.

1. Rectification is determining ____________

a) Length of a line

b) Length of a curve

c) Area of an object

d) Perimeter of an object

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Explanation: Rectification is the mathematical process in which the length of a curve is determined by using integral calculus. Here, the curve is segmented into parts of known length and therefore the length is found.

2. Which one of the following is an infinite curve?

a) Hyperbola

b) Koch curve

c) Gaussian curve

d) Parabola

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Explanation: A curve which has no top limit is the infinite curve. Every arc on the curve has undetermined length. Example: Koch curve.

3. The expression for arc length in rectangular form is_________________

a) \(ds = \int_a^b xy \sqrt{1 + (\frac{dy}{dx})^2} \)

b) \(ds = \int_a^b \sqrt{1 – (\frac{dy}{dx})^2} \,dx \)

c) \(ds = \int_a^b \sqrt{(\frac{dy}{dx})^2} \,dx \)

d) \(ds = \int_a^b \sqrt{1 – (\frac{dy}{dx})^2} \,dx \)

View Answer

Explanation: Rectangular form is represented by x,y and z co-ordinates. The right answer is \(ds = \int_a^b \sqrt{1 – (\frac{dy}{dx})^2} \,dx. \)

4. The expression for arc length in parametric form is_________________

a) \(ds = \int_a^b xy \sqrt{(\frac{dx}{dt})^2 + (\frac{dy}{dt})^2} \,ds \)

b) \(ds = \int_a^b \sqrt{(\frac{dx}{dt})^2 + (\frac{dy}{dt})^2} \,dt \)

c) \(ds = \int_a^b \sqrt{(\frac{dx}{dt})^2 + (\frac{dy}{dt})^2} \,dx \)

d) \(ds = \int_a^b \sqrt{(\frac{dx}{dt})^2 + (\frac{dy}{dt})^2} \)

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Explanation: The parametric equation has more than one dependent variable. The right answer is \(ds = \int_a^b \sqrt{(\frac{dx}{dt})^2 + (\frac{dy}{dt})^2} \,dt \).

5. The expression for arc length in polar form is_________________

a) \(ds = \int_a^b \sqrt{r^2 + (\frac{dr}{dθ})^2} \)

b) \(ds = \int_a^b \sqrt{r^2 + (\frac{dr}{dθ})^2} \,dθ \)

c) \(ds = \int_a^b \sqrt{(\frac{dr}{dθ})^2} \,dθ \)

d) \(ds = \int_a^b \sqrt{r^2 + (\frac{dr}{dθ})^2} \,dr \)

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Explanation: Polar equation is written in terms of polar coordinates r and θ. The right answer is \(ds = \int_a^b \sqrt{r^2 + (\frac{dr}{dθ})^2} \,dθ \).

6. Closed form solutions are absent for ellipses.

a) True

b) False

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Explanation: Closed form solutions are arrived when only a known number of operations are involved. But in the case of ellipse, it is not obtained.

7. What is the length of a circular curve when θ is in degrees and ‘r’ is the radius?

a) s = rθ

b) s = r

c) \(s = \frac{πrθ}{180} \)

d) s = πr

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Explanation: Degrees must be converted to equivalent radians and therefore the arc length is found considering both the radius and angle.

8. The length of curve r = e^{θ}, θ value ranges between 0 to π is________

a) \(\sqrt{2} \)

b) \(\sqrt{2} \,(e^2π – 1) \)

c) \(\sqrt{2} \,(e^2π + 1) \)

d) \(\sqrt{2} \,(e^2π) \)

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Explanation: \(s = \int_a^b \sqrt{r^2 + (\frac{dr}{dθ})^2} \,dθ \)

\(= \int_0^2π \sqrt{(e^θ)^2 + (e^θ)^2)} \,dθ \)

\(= \int_0^2π e^θ \sqrt{2} \,dθ \)

\(= (e^θ \sqrt{2}) |_0^{2π} \)

\(= \sqrt{2} (e^{2π}-1). \)

9. The arc length of y = coshx where x varies from 0 to 1 is ____________

a) \(\frac{e}{2} – \frac{1}{2e} \)

b) \(\frac{e}{2} – \frac{1}{2e} + \frac{1}{2e} \)

c) \(\frac{e}{2} – \frac{1}{2e} \)

d) 2e

View Answer

Explanation: y’ = sinhx

Arc length = \(\int_a^b \sqrt{1 + (\frac{dy}{dx})^2} \,dx\)

= \(\int_0^1 \sqrt{1 + (sinhx)^2)} \,dx\)

= \(\int_0^1 coshx \,dx \)

= \(\Big[sinh x\Big]_0^1\)

= \(\frac{e}{2} – \frac{1}{2e} \).

**Sanfoundry Global Education & Learning Series – Differential and Integral Calculus.**

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