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,\frac14\right)\cup\left(\frac14,\frac12\right)\cup\left(\frac12,\frac34\right)\cup\left(\frac34,1\right)`

with respective midpoints of
`\frac18,\frac38,\frac58,\frac78`. The length of each interval is
`\frac14`. Note that I'm also assuming you are supposed to use equally spaced intervals.


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

`\approx\frac{\frac18-\left(\frac18\left)^3}4+\frac{\frac38-\left(\frac38\left)^3}4+\frac{\frac58-\left(\frac58\left)^3}4+\frac{\frac78-\left(\frac78\left)^3}4=(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)=\frac1{5x}\cdot(\mathrm d(5x))/(\mathrm dx)-\frac1{2x}\cdot(\mathrm d(2x))/(\mathrm dx)`

`F'(x)=\frac5{5x}-\frac2{2x}`

`F'(x)=\frac1x-\frac1x`

`F'(x)=0`

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