Integration

Remember the advice from week 1: if there’s any particular question you find difficult, try using ELM to create some similar questions to practice with before moving on to harder questions.

1 Integrals

  1. Find \(\int f(x)~\text{d}x\) when \(f(x) = x^2\), \(f(x) = 3\), and \(f(x) = x^3 + 3x^2 + 3x + 1\).

  2. Find the area under the previous curves between \(0\) and \(1\).

  3. Find \(\int f(t)~\text{d}t\) when \(f(t) = 2e^{2t}\), \(f(t) = 2\cos(2t)\), and \(f(t) = 1/\cos^2(t)\).

2 Using Parts

The integration by parts formula, \[ \int u(x)v'(x)~\text{d}x = u(x)v(x) - \int u'(x)v(x)~\text{d}x, \] gives us a way to break down complicated integrals that involve products.

  1. Find \(I = \int xe^x~\text{d}x.\)

  2. Find \(I_2 = \int x^2 e^x~\text{d}x.\)

  3. Find \(I_3 = \int x^3 e^x~\text{d}x.\)

  4. Is there a formula for \(I_n = \int x^n e^x~\text{d}x\), where \(n\) is a strictly positive integer?

3 Areas

A shape \(D_n\) is defined as the space enclosed by the four line segments \[\begin{equation*} \begin{matrix} y = h\left(x-b\right)^n,\ 0\leq x \leq b & \quad & y = h\left(x+b\right)^n,\ -b\leq x\leq 0 \\ y = -h\left(x+b\right)^n,\ -b\leq x\leq 0 & \quad & y = -h\left(x-b\right)^n,\ 0\leq x\leq b \end{matrix} \end{equation*}\] where \(b\) and \(h\) are positive real numbers and \(n\) is a positive integer.

  1. What is the area of \(D_1\)?
Hint

What does \(D_1\) look like?

  1. What is the area of \(D_2\)?

  2. What role do the domains of \(b\), \(h\), and \(n\) play?