Question

HELP ME PASS CALCULUS CLASS PLS - THERE IS A PART ONE WITH AN INAPPROPRIATE ANSWER PLS HELP ME

Answer 1

For the first question, the region is a bit ambiguous. $x$ and $x^3$ intersect three times, and there are two regions between them. So either you're approximating

$\displaystyle\int_{-1}^1|x^3-x|\,\mathrm dx$

or

$\displaystyle\int_0^1(x-x^3)\,\mathrm dx$

I'll assume the second case. Split the interval into 4 smaller ones, taking

$(0,1)=\left(0,\dfrac14\right)\cup\left(\dfrac14,\dfrac12\right)\cup\left(\dfrac12,\dfrac34\right)\cup\left(\dfrac34,1\right)$

with respective midpoints of $\dfrac18,\dfrac38,\dfrac58,\dfrac78$. The length of each interval is $\dfrac14$. Note that I'm also assuming you are supposed to use equally spaced intervals.

$\displaystyle\int_0^1(x-x^3)\,\mathrm dx$
$\approx\dfrac{\frac18-\left(\frac18\left)^3}4+\dfrac{\frac38-\left(\frac38\left)^3}4+\dfrac{\frac58-\left(\frac58\left)^3}4+\dfrac{\frac78-\left(\frac78\left)^3}4=\dfrac{33}{128}$

Skipping the second one since I already answered it.

For the third, split up the region of integration at some arbitrary constant $c$ between $2x$ and $5x$, then differentiate and apply the fundamental theorem of calculus.

$F(x)=\displaystyle\int_{2x}^{5x}\frac{\mathrm dt}t$
$F(x)=\displaystyle\int_c^{5x}\frac{\mathrm dt}t+\int_{2x}^c\frac{\mathrm dt}t$
$F(x)=\displaystyle\int_c^{5x}\frac{\mathrm dt}t-\int_c^{2x}\frac{\mathrm dt}t$

$F'(x)=\dfrac1{5x}\cdot\dfrac{\mathrm d(5x)}{\mathrm dx}-\dfrac1{2x}\cdot\dfrac{\mathrm d(2x)}{\mathrm dx}$
$F'(x)=\dfrac5{5x}-\dfrac2{2x}$
$F'(x)=\dfrac1x-\dfrac1x$
$F'(x)=0$

Since $F'(x)=0$, it follows that $F(x)$ is constant.