What's the 3D shape you should be afraid of?

AP CALC QUESTION!! WILL MARK BRAINLIEST (TWO QUESTIONS)

kodGreya [7K]

1. Using washers, the volume is given by the integral

$\displaystyle\pi\int_1^{e^2}((2+2)^2-(\ln x+2)^2)\,\mathrm dx$

$=\boxed{\displaystyle\pi\int_1^{e^2}(20+4\ln x-(\ln x)^2)\,\mathrm dx}$

We're using washers whose centers depend on the value of $x$ , hence we integrate with respect to

2. The area of the given region is given by the integral

$\displaystyle\int_0^2\sin^{-1}\frac x2\,\mathrm dx$

To compute the integral, first consider the substitution $u=\frac x2$ , or $2u=x$ so that $2\,\mathrm du=\mathrm dx$ . Then $x\to0\implies u\to0$ and $x\to2\implies u\to1$ , so the integral is equivalently

$\displaystyle2\int_0^1\sin^{-1}u\,\mathrm du$

Integrate by parts, taking

$f=\sin^{-1}u\implies\mathrm df=\dfrac{\mathrm du}{\sqrt{1-u^2}}$

$\mathrm dg=\mathrm du\implies g=u$

so that

$\displaystyle2\int_0^1\sin^{-1}u\,\mathrm du=2\left(u\sin^{-1}u\bigg|_0^1-\int_0^1\frac u{\sqrt{1-u^2}}\,\mathrm du\right)$

$\sin^{-1}0=0$ and $\sin^{-1}1=\frac\pi2$ , so the area is

$\displaystyle\pi-2\int_0^1\frac u{\sqrt{1-u^2}}\,\mathrm du$

For the remaining integral, substitute $w=1-u^2$ , so that $\mathrm dw=-2u\,\mathrm du$ . Then $u\to0\implies w\to1$ and $u\to1\implies w\to0$ :

$\displaystyle\pi-\int_0^1\frac{\mathrm dw}{\sqrt w}$

(notice that the integral is improper)

$\displaystyle\pi-\lim_{t\to0^+}2\sqrt w\bigg|_t^1$

$\displaystyle\pi-2\left(1-\lim_{t\to0^+}\sqrt t\right)=\boxed{\pi-2}$

8 0

3 years ago

Since Emma beat his score on the test, Dean decided he was having trouble in his AP Psychology class. Despite his efforts and re

tatyana61 [14]

PSYCHOANALYTIC:

Focuses on the unconscious motives of the person in terms of Id, Ego, and Superego.

Therapy may involve hyponesis, free association, and dream analysis.

HUMANISTIC:

Focuses on the value, dignity, and worth of each person.

Therapy may involve non-directive therapy, active listening, unconditional positive regard, such as calling your patient a "client" to show equality.

COGNITIVE: (does not include social)

Focuses on the way people think.

Therapy may involve rational-emotive therapy, which helps change unrealistic assumptions of onself,

I couldn't find social, but if it's similar, I have a section on:

BEHAVIOR:

Focuses more on a person's actual behavior rather than their thoughts.

Therapy may involve counter-conditioning, which is pairing a stimulus (positive thing) with good behavior, to encourage that behavior more often.

The following was taken out of my Psychology 1 course notes, enjoy!

8 0

3 years ago

Explain how the facts in Shaw v. Reno (1993) led to a different holding than in League of United Latin American Citizens v. Perr

Pachacha [2.7K]

Jsjsjkajamskssjnsksisjejwjwjsnsksjsmskskjsks

8 0

3 years ago

Add on to this plz❤️Or restart for social

Inessa05 [86]

Kinda confused but only did it for points sooo

5 0

3 years ago

What is the axis of symetry of the function f(x)= -(x+9)(x-2)

Stels [109]

We first need to write the function in standard form.

y = -x^2 - 7x + 18

To find the axis of symmetry, we do -b/2a

The a and b are the coefficients of the x values.

So a would be -1 since -x is same as -1x

Follow that, b would be -7.

Then just plug the values in. - (-7)/2(-1) = -7/2

The axis of symmetry is -7/2

5 0

3 years ago