Mathematics, 09.07.2021 05:50 Deemon

Find the area of the region between the curve x^3+2x^2-3x and the x-axis over the interval [-3,1]

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Answer from: serenityharmon1

$\displaystyle A = \frac{32}{3}$

General Formulas and Concepts:

Calculus

Integrals

Definite IntegralsArea under the curve

Integration Rule [Reverse Power Rule]: $\displaystyle \int {x^n} \, dx = \frac{x^{n + 1}}{n + 1} + C$

Integration Rule [Fundamental Theorem of Calculus 1]: $\displaystyle \int\limits^b_a {f(x)} \, dx = F(b) - F(a)$

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

Integration Property [Addition/Subtraction]: $\displaystyle \int {[f(x) \pm g(x)]} \, dx = \int {f(x)} \, dx \pm \int {g(x)} \, dx$

Area of a Region Formula: $\displaystyle A = \int\limits^b_a {[f(x) - g(x)]} \, dx$

Step-by-step explanation:

Step 1: Define

Identify

Curve: x³ + 2x² - 3x

Interval: [-3, 1]

Step 2: Find Area

Set up: $\displaystyle A = \int\limits^1_{-3} {(x^3 + 2x^2 - 3x)} \, dx$ [Integral] Rewrite [Integration Property - Addition/Subtraction]: $\displaystyle A = \int\limits^1_{-3} {x^3} \, dx + \int\limits^1_{-3} {2x^2} \, dx - \int\limits^1_{-3} {3x} \, dx$ [Integrals] Rewrite [Integration Property - Multiplied Constant]: $\displaystyle A = \int\limits^1_{-3} {x^3} \, dx + 2\int\limits^1_{-3} {x^2} \, dx - 3\int\limits^1_{-3} {x} \, dx$ [Integrals] Integrate [Integration Rule - Reverse Power Rule]: $\displaystyle A = (\frac{x^4}{4}) \bigg| \limits^1_{-3} + 2(\frac{x^3}{3}) \bigg| \limits^1_{-3} - 3(\frac{x^2}{2}) \bigg| \limits^1_{-3}$ Evaluate [Integration Rule - Fundamental Theorem of Calculus 1]: $\displaystyle A = -20 + 2(\frac{28}{3}) - 3(-4)$ Evaluate: $\displaystyle A = \frac{32}{3}$

Topic: AP Calculus AB/BC (Calculus I/I + II)

Unit: Integration

Book: College Calculus 10e

Answer from: Quest

ha!

i just did this!

18. g

20. h 30 degree turn.

21. d 90 degree angles

and also 19 is a. a sphere not a cirlcle

hope i !

pls mark me brainiest

Answer from: Quest

The answer is 26 to 1

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Find the area of the region between the curve x^3+2x^2-3x and the x-axis over the interval [-3,1]...

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