# Differential and Integral Calculus Questions and Answers – Rectification

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

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

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

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

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

Explanation: Polar equation is written in terms of polar coordinates r and θ. The right answer is $$ds = \int_a^b \sqrt{r^2 + (\frac{dr}{dθ})^2} \,dθ$$.
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6. Closed form solutions are absent for ellipses.
a) True
b) False

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

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 = eθ, θ value ranges between 0 to π is________
a) $$\sqrt{2}$$
b) $$\sqrt{2} \,(e^2π – 1)$$
c) $$\sqrt{2} \,(e^2π + 1)$$
d) $$\sqrt{2} \,(e^2π)$$

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 = cosh⁡x 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

Explanation: y’ = sinh⁡x
Arc length = $$\int_a^b \sqrt{1 + (\frac{dy}{dx})^2} \,dx$$
= $$\int_0^1 \sqrt{1 + (sinh⁡x)^2)} \,dx$$
= $$\int_0^1 cosh⁡x \,dx$$
= $$\Big[sinh ⁡x\Big]_0^1$$
= $$\frac{e}{2} – \frac{1}{2e}$$.

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