Find the ratio of shaded portions to unshaded portions in fig 8.1

Solve the Anti derivative.

Alex Ar [27]

Answer:

$\displaystyle \int {\frac{1}{9x^2+4}} \, dx = \frac{1}{6}arctan(\frac{3x}{2}) + C$

General Formulas and Concepts:

Algebra I

Factoring

Calculus

Antiderivatives - integrals/Integration

Integration Constant C

U-Substitution

Integration Property [Multiplied Constant]: $\displaystyle \int {cf(x)} \, dx = c \int {f(x)} \, dx$

Trig Integration: $\displaystyle \int {\frac{du}{a^2 + u^2}} = \frac{1}{a}arctan(\frac{u}{a}) + C$

Step-by-step explanation:

Step 1: Define

$\displaystyle \int {\frac{1}{9x^2 + 4}} \, dx$

Step 2: Integrate Pt. 1

[Integral] Factor fraction denominator: $\displaystyle \int {\frac{1}{9(x^2 + \frac{4}{9})}} \, dx$
[Integral] Integration Property - Multiplied Constant: $\displaystyle \frac{1}{9} \int {\frac{1}{x^2 + \frac{4}{9}}} \, dx$

Step 3: Identify Variables

Set up u-substitution for the arctan trig integration.

$\displaystyle u = x \\ a = \frac{2}{3} \\ du = dx$

Step 4: Integrate Pt. 2

[Integral] Substitute u-du: $\displaystyle \frac{1}{9} \int {\frac{1}{u^2 + (\frac{2}{3})^2} \, du$
[Integral] Trig Integration: $\displaystyle \frac{1}{9}[\frac{1}{\frac{2}{3}}arctan(\frac{u}{\frac{2}{3}})] + C$
[Integral] Simplify: $\displaystyle \frac{1}{9}[\frac{3}{2}arctan(\frac{3u}{2})] + C$
[integral] Multiply: $\displaystyle \frac{1}{6}arctan(\frac{3u}{2}) + C$
[Integral] Back-Substitute: $\displaystyle \frac{1}{6}arctan(\frac{3x}{2}) + C$

Topic: AP Calculus AB

Unit: Integrals - Arctrig

Book: College Calculus 10e

7 0

3 years ago

Find the ratio of the width to the length of the rectangle, then simplify the ratio. Use the conversion 100 cm = 1 m.

valkas [14]

The answer to the question is c

3 0

3 years ago

Solve everything for brainiest

IRISSAK [1]

Answer:

1. $a = 8$

2. $b = \frac{1}{8}$

3. $c = 2 \frac{3}{5}$

4. $d = \frac{13}{50}$

Step-by-step explanation:

6 0

3 years ago

the number of grams a woman should eat in a day to maintain her weight is proportional to her weight. a woman weighing 102 pound

vitfil [10]

Given:
grams of fat : 34 grams TO weight of woman : 102 pounds
grams of fat : ? TO weight of woman : 180 pounds

This is a proportion problem: 34 grams to 102 pounds.

We first have to convert a unit of measure to another to maintain uniformity of measure. let us convert pounds to grams:

102 pounds * 453.592 grams / pound = 46,266.384 grams
180 pounds * 453.592 grams / pound = 81,646.560 grams

34 grams to 46,266.384 grams = x grams to 81,646.560 grams

proportion: a:b = c:d where ad = bc

34 grams * 81,646.560 grams = x * 46,266.384 grams
2,775,983.04 grams² = x * 46,266.384 grams
2,775,983.04 grams² ÷ 46,266.384 grams = x
60 grams = x

A woman weighing 180 pounds should eat 60 grams of fat to maintain her weight.

7 0

3 years ago

It is found that 5 out of every 8 college students like algebra. If a certain college has 4,000 students, how many of them like

solniwko [45]

5 out of 8 turned into a fraction is 5/8. 5/8 turned into a percentage is 62.5%. 62.5% of 4,000 is 2,500. 2,500 students in the college like algebra.
Hope this helped.

3 0

4 years ago

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Find the ratio of shaded portions to unshaded portions in fig 8.1​

Find the ratio of shaded portions to unshaded portions in fig 8.1