Symbolic Integration Assignment.sagews

Symbolic Integration Assignment

Question 0

Watch the lecture video here.

Did you watch the video? [Type yes or no.]

Question 1

Compute the following integrals using Sage.

Part a

$\displaystyle\int \sin(3x)\sin(2x)\, dx$

Part b

$\displaystyle\int e^{5t}\sin(4t)\, dt$

Part c

$\displaystyle\int_0^{\pi/2} \sin(ax)^2\, dx$

Part d

$\displaystyle\int_1^5\frac{\ln(x)}{x^2}\, dx$

Part e

$\displaystyle\int_0^1 x\tan(x)\, dx$

[Use numerical_integral]

Question 2

The velocity at time $t$ of a particle travelling in a straight line is given by the equation $v(t)=3t^3-4t^2+10$. How far does the particle travel from $t=10$ to $t=20$?

[Hint: Distance traveled is the integral of velocity.]

Question 3

Let $f(x)=2x\sqrt{1-x^3}$.

Part a

Find the area between the graph of $f$ and the x-axis from $x=0$ to $x=1$. Convert Sage's answer to a decimal.

Part b

Estimate the area in Part a using left and right Riemann sums with $n=100$ subintervals.

Question 4

Use Sage to calculate $\displaystyle\frac{d}{dx}\int_{x}^{\sin(x)}3t^2\,dt$.

Note: The Fundamental Theorem of Calculus implies that $\displaystyle\frac{d}{dx}\int_{g(x)}^{h(x)}f(t)\,dt=f(h(x))h'(x)-f(g(x))g'(x)$.

Question 5

Use Sage to calculate $\displaystyle\int_5^{10}\frac{d}{dx}\frac{5}{1-x^2}\,dx$.

Note: The Fundamental Theorem of Calculus implies that $\displaystyle\int_a^{b}\frac{d}{dx}f(x)\,dx=f(b)-f(a)$.

This material was developed by Aaron Tresham at the University of Hawaii at Hilo and is
licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.