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

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

[tex]\displaystyle\int_{-1}^1|x^3-x|\,\mathrm dx[/tex]

or

[tex]\displaystyle\int_0^1(x-x^3)\,\mathrm dx[/tex]

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

[tex](0,1)=\left(0,\dfrac14\right)\cup\left(\dfrac14,\dfrac12\right)\cup\left(\dfrac12,\dfrac34\right)\cup\left(\dfrac34,1\right)[/tex]

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

[tex]\displaystyle\int_0^1(x-x^3)\,\mathrm dx[/tex]
[tex]\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}[/tex]

Skipping the second one since I already answered it.

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

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

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

Since [tex]F'(x)=0[/tex], it follows that [tex]F(x)[/tex] is constant.