We will cover following topics

Definite Integrals

Riemanns’ Definition of Definite Integrals

The definite integral of a continuous function $f$ over the interval $[a, b]$ is the limit of a Riemann sum as the number of subdivisions approach infinity. Mathematically,

\int_{a}^{b} f (x) d x = lim_{n \to \infty} \sum_{i = 1}^{n} Δ x \cdot f (x_{i})

where $Δ x = \frac{b - a}{n}$ , and $x_{i} = a + Δ x \cdot i$

Indefinite Integrals

An integral of the form $\int f (z) d z$ , without lower and upper limits, is called an indefinite integral.

It is also called an antiderivative.

Infinite Integrals

These occur when the limits of integration or the integrand become infinite.

Improper Integrals

If the interval over which the definite integral $\int_{a}^{b} f (x) d x$ is defined is infinite or if the function $f (x)$ has a discontinuity in $[a, b]$ or both of these conditions are satisfied, the integral is called an improper integral.

Observations:

The improper integral $\int_{1}^{\infty} \frac{1}{x^{p}} d x$ converges when $p > 1$ and diverges when $p \leq 1$ $.
The improper integral $\int_{0}^{1} \frac{1}{x^{p}} d x$ converges when $p < 1$ and diverges when $p \geq 1$ .

Direct Comparison Test for Improper Integrals: Let $f$ and $g$ be continuous on $[a, \infty)$ where $0 \leq f (x) \leq g (x)$ for all $x$ in $[a, \infty)$ .

If $\int_{a}^{\infty} g (x) d x$ converges, then $\int_{a}^{\infty} f (x) d x$ converges.
If $\int_{a}^{\infty} f (x) d x$ diverges, then $\int_{a}^{\infty} g (x) d x$ diverges.

Limit Comparison Test for Improper Integrals: Let $f$ and $g$ be continuous functions on $[a, \infty)$ where $f (x) > 0$ and $g (x) > 0$ for all $x$ . If

lim_{x \to \infty} \frac{f (x)}{g (x)} = L, 0 < L < \infty

then $\int_{a}^{\infty} f (x) d x$ and $\int_{a}^{\infty} g (x) d x$ either both converge or both diverge.

Double Integrals

Consider a function of two variables $f (x, y)$ . The definite integral denoted by $\iint_{R} f (x, y) d A$ , where $R$ is the region of integration in $x y - p l a n e$ , is called as a double inetegral.

For positive $f (x, y)$ , this definite integral gives the volume under the surface $f (x, y)$ and above $x y - p l a n e$ , for $x$ and $y$ in the region $R$ .

Triple Integrals

The integration of a function $f (x, y, z)$ over a three-dimensional region $R$ in $x y z - s p a c e$ is called a triple integral and is denoted as:

∭_{R} f (x, y, z) d V

PYQs

Definite Integrals

1) Evaluate the following integral:

\int_{π / 6}^{π / 3} \frac{\sqrt[3]{\sin x}}{\sqrt[3]{\sin x} + \sqrt[3]{\cos x}} d x

[2015, 10M]

2) Evaluate $\int_{0}^{1} \frac{\log_{e} (1 + x)}{1 + x^{2}} d x$

[2014, 10M]

Infinite Integrals

1) Show that $e^{- x} x^{n}$ is bounded on $[0, \infty)$ for all positive integral values of $n$ . Using this result, show that $\int_{0}^{\infty} e^{- x} x^{n} d x$ exists.

[2007, 25M]

2) Show that $\int_{0}^{\infty} \int_{0}^{\infty} e^{- (x^{2} + y^{2})} d x d y = \frac{π}{4}$

[2002, 12M]

Improper Integrals

1) Examine if the improper integral $\int_{0}^{3} \frac{2 x d x}{{(1 - x^{2})}^{2 / 3}}$ exists.

[2017, 10M]

2) Evaluate $I = \int_{0}^{1} \sqrt[3]{x \log (\frac{1}{x})} d x$

[2016, 10M]

3) Evaluate $\int_{0}^{1} (2 x \sin \frac{1}{x} - \cos \frac{1}{x}) d x$

[2013, 10M]

4) Find all the real values of $p$ and $q$ so that the integral $\int_{0}^{1} x^{p} {(\log \frac{1}{x})}^{q} d x$ converges.

[2012, 20M]

5) Does the integral $\int_{- 1}^{1} \sqrt{\frac{1 + x}{1 - x}} d x$ exist? If so, find its value.

[2010, 12M]

6) Evaluate $\int_{0}^{1} (x ℓ n x)^{3} d x$

[2008, 12M]

7) Prove that the function $f$ defined on $[0, 4]$ by $f (x) = [x]$ , the greatest integer $\leq x, x \in [0, 4]$ is integrable on $[0, 4]$ and that $\int_{0}^{4} f (x) d x = 6$ .

[2004, 12M]

8) Test the convergent of the integrals:-
i) $\int_{0}^{1} \frac{d x}{x^{1 / 3} (1 + x^{2})}$
ii) $\int_{0}^{\infty} \frac{\sin^{2} x}{x^{2}} d x$

[2003, 15M]

9) Test the convergence of $\int_{0}^{1} \frac{\sin (\frac{1}{x})}{\sqrt{x}} d x$ .

[2001, 12M]

Double Integrals

1) Evaluate the integral $\int_{0}^{a} \int_{x / a}^{x} \frac{x d y d x}{x^{2} + y^{2}}$ .

[2018, 12M]

2) Integrate the function $f (x, y) = x y (x^{2} + y^{2})$ over the domain $R : {- 3 \leq x^{2} - y^{2} \leq 3, 1 \leq x y \leq 4}$

[2017, 10M]

3) Prove that $\frac{π}{3} \leq \iint \frac{d x d y}{\sqrt{x^{2} + (y - 2)^{2}}} \leq π$ , where $D$ is the unit disc.

[2017, 10M]

4) Evaluate $\iint_{R} f (x, y) d x d y$ over the rectangle $R = [0, 1; 0, 1]$ , where $f (x, y) = {\begin{matrix} x + y, if x^{2} < y < 2 x^{2} \\ 0, \end{matrix}$

[2016, 15M]

5) Evaluate the integral $\iint (x - y)^{2} \cos^{2} (x + y) d x d y$ where $R$ is the rhombus with successive vertices as $(π, 0)$ , $(2 π, π)$ , $(π, 2 π)$ , $(0, π)$ .

[2015, 12M]

6) Evaluate $\iint_{R} \sqrt{| y - x^{2} |} d x d y$ where $R = [- 1, 1; 0, 2]$ .

[2015, 13M]

7) Evaluate $\iint_{D} x y d A$ , where $D$ is the region bounded by the line $y = x - 1$ and the parabola $y^{2} = 2 x + 6$ .

[2013, 15M]

8) Show that the function

f (x) = [x^{2}] + | x - 1 |

is Riemann integrable m the interval [0, 2], where $[α]$ denotes the greatest integer less than or equal to $α$ . Can you give an example of a function that is not Riemann integrable on [0, 2]? Compute $\int_{0}^{2} f (x) d x$ , where $f (x)$ is as above.

[2010, 12M]

9) Evaluate $\int_{0}^{1} \ln x d x$ .

[2011, 12M]

10) Evaluate $I = \iint_{s} (x d y d z + d z d x + x z^{2}) d x d y$ , where $S$ is the outer side of the part of the sphere $x^{2} + y^{2} + z^{2} = 1$ in the first octant.

[2009, 20M]

11) Evaluate the double integral $\iint_{y}^{a} \frac{x d x d y}{x^{2} + y^{2}}$ by changing the order of integration.

[2008, 20M]

12) Change the order of integration in $\int_{0}^{\infty} \int_{x}^{\infty} \frac{e^{- y}}{y} d y d x$ and hence evaluate it.

[2006, 15M]

13) Find the x-coordinate of the centre of gravity of the solid lying inside the cylinder $x^{2} + y^{2} = 2 a x$ , between the plane $z = 0$ and the paraboloid $x^{2} + y^{2} = a z$ .

[2005, 15M]

14) Find the centre of gravity of the region bounded by the curve ${(\frac{x}{a})}^{2 / 3} + {(\frac{x}{b})}^{2 / 3} = 1$ and both axes in the first quadrant, the density being $ρ = k x y$ , where $k$ is a constant.

[2002, 15M]

Triple Integrals

1) Let $D$ be the region determine by the inequalities $x > 0, y > 0, z < 8$ and $z > x^{2} + y^{2}$ . Compute $∭_{D} 2 x d x d y d z$ .

[2010, 20M]

2) Evaluate $∭ (x + y + z + 1)^{2} d x d y d z$ over the region defined by $x \geq 0$ , $y \geq 0$ , $z \geq 0$ , $x + y + z \leq 1$ .

[2001, 15M]

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