Mathematics, 29.09.2019 00:30, wallacelizzy1

How would you use the fundamental theorem of calculus to determine the value(s) of b if the area under the graph g(x)=4x between x=1 and x=b is equal to 240?

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answered: maddie53116

$\displaystyle b = 11$

General Formulas and Concepts:

Pre-Algebra

Order of Operations: BPEMDAS

Brackets Parenthesis Exponents Multiplication Division Addition Subtraction Left to Right

Equality Properties

Multiplication Property of Equality Division Property of Equality Addition Property of Equality Subtraction Property of Equality

Algebra I

Functions

Function Notation

Calculus

Integration

IntegralsDefinite IntegralsIntegration Constant C

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]: $\displaystyle \int {cf(x)} \, dx = c \int {f(x)} \, dx$

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

Step-by-step explanation:

Step 1: Define

Identify

g(x) = 4x

Interval [1, b]

A = 240

Step 2: Solve for b

Substitute in variables [Area of a Region Formula]: $\displaystyle \int\limits^b_1 {4x} \, dx = 240$ [Integral] Rewrite [Integration Property - Multiplied Constant]: $\displaystyle 4\int\limits^b_1 {x} \, dx = 240$ [Integral] Integrate [Integration Rule - Reverse Power Rule]: $\displaystyle 4(\frac{x^2}{2}) \bigg| \limits^b_1 = 240$ Evaluate [Integration Rule - Fundamental Theorem of Calculus 1]: $\displaystyle 4(\frac{b^2}{2} - \frac{1}{2}) = 240$ [Distributive Property] Distribute 4: $\displaystyle 2b^2 - 2 = 240$ [Addition Property of Equality] Add 2 on both sides: $\displaystyle 2b^2 = 242$ [Division Property of Equality] Divide 2 on both sides: $\displaystyle b^2 = 121$ [Equality Property] Square root both sides: $\displaystyle b = \pm 11$ Choose: $\displaystyle b = 11$

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

Unit: Integration

Book: College Calculus 10e

answered: Guest

mx+89b(-98)nb

step-by-step explanation:

answered: Guest

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How would you use the fundamental theorem of calculus to determine the value(s) of b if the area und...