# MT #5: Integral Calculus

#### Timings: 10:30-13:30 Hrs Date: 21 March 2021

## Part A: Short-answer type questions

**Submission file:** Write answers to all the ten questions on a single sheet of paper. Email a picture of your answer sheet. Name the file as PartA.jpg.

**For this part, answers must be written without any explanation.**

- Suppose \(n\) is a fixed positive integer. If \( \int_{1}^{k} x^{n-1} dx = \frac{1}{n} \), what is the value of \(k\)?
- Evaluate \( \displaystyle \int_{0}^{2} \frac{1}{\left( x-\frac{1}{2} \right)^2+\frac{3}{4}} dx \).
- Let \( f(x) \) be a continuous real-valued function on \([0, 1]\) for which \[ \int_{0}^{1} f(x)dx = 0 \text{ and }\;\;\; \int_{0}^{1} xf(x) dx = 1 \] Give an example of such a function.
- Consider the integral \( \int_{0}^{\infty} \frac{ \sin^2 x }{\pi^2 - x^2} dx \). Which of these statements are true?

(a) The integral does not exist as the function is not defined at \( x=\pi \).

(b) There is a removable singularity at \(x=\pi\) but the integral does not exist.

(c) There is a removable singularity at \(x=\pi\) and the integral exists. - Find the volume of the solid obtained when the region bounded by \( y=\sqrt{2x}\), \( y=-x \) and the line \(x=6\) is revolved around the x-axis.
- Let \( f(x) = \sum_{n=0}^{\infty} \frac{x2^n}{n!} \) be a function defined in the interval \( [0,10] \). Find the value of: \[ \int_{0}^{10} f(x) dx \]
- For any real number \(x\), let \([x]\) denote the greatest integer \(m\) such that \(m \leq x\). Find the value of \( \int_{-2}^{2} [x^2 – 2]dx \).
- What is the area bounded by the curve \( y = \ln x \), the x-axis and the straight line \(x=3\)?
- Suppose \(f(x)\) is a double differentiable function with \(f^{\prime}(0) = 1\) and \(f^{\prime}(1) = 2\). What is the value of the integral below? If it possible for the integral to take multiple values, write
**Indeterminate**. \[ \int_{0}^{1} f^\prime(x) f^{\prime\prime}(x) dx \] - Compute the limit: \[ \lim_{n\rightarrow\infty} n^3 \int_{0}^{1/n} x^{100x+2} dx \]

## Part B: Subjective questions

**Submission files:** Each question in this part must be answered on a page of its own. Name the files as B1.jpg, B2.jpg, etc. In case you have multiple files for the same question, say B4, name the corresponding files as B4-1.jpg, B4-2.jpg, etc.

**Clearly explain your entire reasoning.** No credit will be given without reasoning. Partial solutions may get partial credit.

**B1.** Let \( f:[0,1] \rightarrow \mathbb{R}^+ \) be a continous function such that \( \int_{0}^{1} f(t) dt = \frac{1}{3} \). Show that there exists \( c\in(0,1) \) such that \( \int_{0}^{c} f(t) dt = c-\frac{1}{2}\).

**B2.** Let \( f(x) \) be a real-valued function defined on the interval \( [1,\infty) \). The following equations hold: \begin{align*} f(1) &= 1 \\ f^{\prime}(x) &= \frac{1}{x^2+f(x)^2} \end{align*}

Prove that \( \lim_{x\rightarrow \infty} f(x) < 1 + \pi/4 \).

**B3.** Consider the parabola given by \(y = x^2\). The normal is constructed at a point \(P\) and meets the parabola again in \(Q\). Determine the coordinate of \(P\) for which the arc length along the parabola between \(P\) and \(Q\) is minimized.

**B4.** Let \( f(x) \) be a continous function defined on the interval \(I=[0,1]\) with the property \[ yf(x) + xf(y) \leq 1 \] for \(x,y\) in \(I\).

(a) Prove that: \[ \int_{0}^{1} f(x) dx \leq \frac{\pi}{4} \] (b) Find a function \( f(x) \) for which equality holds in problem (a).

**B5**. Suppose \(\displaystyle I_{n}= \int_{0}^{2} (2x-x^2)^{n}dx\). Prove that \(\displaystyle \lim_{n\rightarrow \infty} I_{n} = 0 \).

**B6**. Let \( \displaystyle f(x) = \frac{x}{1+x^5\sin^2x} \). Prove that \(\displaystyle \int_0^\infty f(x)\; dx\) is bounded.