Symbolic Integration Assignment.sagews

Symbolic Integration Assignment

Question 0

[1 point] Watch the lecture video here.

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

Yes

Question 1

[4 points] Compute the following integrals using the integral command.

Part a

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

integral(sin(3*x)*sin(2*x),x)
show(_)

-1/10*sin(5*x) + 1/2*sin(x)

$\displaystyle -\frac{1}{10} \, \sin\left(5 \, x\right) + \frac{1}{2} \, \sin\left(x\right)$

$\displaystyle -\frac{1}{10} \, \sin\left(5 \, x\right) + \frac{1}{2} \, \sin\left(x\right)$

Part b

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

%var t
integral(e^(5*t)*sin(4*t),t)
show(_)

-1/41*(4*cos(4*t) - 5*sin(4*t))*e^(5*t)

$\displaystyle -\frac{1}{41} \, {\left(4 \, \cos\left(4 \, t\right) - 5 \, \sin\left(4 \, t\right)\right)} e^{\left(5 \, t\right)}$

Part c

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

%var a
integral((sin(a*x))^2,x,0,pi/2)
show(_)

1/4*(pi*a - sin(pi*a))/a

$\displaystyle \frac{\pi a - \sin\left(\pi a\right)}{4 \, a}$

Part d

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

integral(ln(x)/x^2,x,1,5)
show(_)

-1/5*log(5) + 4/5

$\displaystyle -\frac{1}{5} \, \log\left(5\right) + \frac{4}{5}$

Question 2

[1 point] Use the numerical_integral command to compute $\displaystyle\int_0^1 x\tan(x)\, dx$

numerical_integral(x*tan(x),0,1)

(0.42808830136517595, 4.7527348874829114e-15)

Question 3

[1 point] 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.]

%var t
numerical_integral(3*t^3-4*t^2+10,10,20)

(103266.66666666669, 1.1464903100962451e-09)

Question 4

[1 point] 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.

integral(2*x*sqrt(1-x^3),x,0,1)
show(_)

2/3*beta(2/3, 3/2)

$\displaystyle \frac{2}{3} \, \operatorname{B}\left(\frac{2}{3}, \frac{3}{2}\right)$

N(2/3*beta(2/3, 3/2))

0.739174159779940

Part b

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

f(x)=2*x*sqrt(1-x^3)
a=0
b=1
n=100
dx=(b-a)/n
%var i
LS=sum(f(a+i*dx)*dx,i,0,n-1)
RS=sum(f(a+i*dx)*dx,i,1,n)

Question 5

[1 point] We are going to compute $\displaystyle\frac{d}{dx}\int_{x}^{\sin(x)}3t^2\,dt$ in two ways.

Part a

Use the derivative and integral commands to calculate $\displaystyle\frac{d}{dx}\int_{x}^{\sin(x)}3t^2\,dt$.

%var t
integral(3*t^2,t,x,sin(x))

-x^3 + sin(x)^3

derivative(-x^3 + sin(x)^3,x)
show(_)

3*cos(x)*sin(x)^2 - 3*x^2

$\displaystyle 3 \, \cos\left(x\right) \sin\left(x\right)^{2} - 3 \, x^{2}$

Part b

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

With $f(t)=3t^2$, $h(x)=\sin(x)$, and $g(x)=x$, calculate $f(h(x))\cdot h'(x)-f(g(x))\cdot g'(x)$. [You should get the same answer as part a.]

Question 6

[1 point] We are going to compute $\displaystyle\int_5^{10}\frac{d}{dx}\frac{5}{1-x^2}\,dx$ in two ways.

Part a

Use the integral and derivative commands to calculate $\displaystyle\int_5^{10}\frac{d}{dx}\frac{5}{1-x^2}\,dx$.

integral(5/1-x^2,x,5,10)

-800/3

derivative(-800/3,x)
show(_)

0

$\displaystyle 0$

Part b

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

With $f(x)=\displaystyle\frac{5}{1-x^2}$, calculate $f(10)-f(5)$. [You should get the same answer as part a.]

f(x)=5/(1-x^2)
integral(f(10-f(5),x,a,b)

Error in lines 2-2 Traceback (most recent call last): File "/cocalc/lib/python3.9/site-packages/smc_sagews/sage_server.py", line 1231, in execute compile(block + '\n', File "<string>", line 1 integral(f(Integer(10)-f(Integer(5)),x,a,b) ^ SyntaxError: unexpected EOF while parsing

%var b
f(x)=5/(1-x^2)
integral(f(10)-f(5),x,a,b);

125/792*b