Mathematics, 28.02.2021 02:10 Becky323

AP Calculus. The answer key says it's A) but I don't how to solve it.

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Answer from: yoogirlchloee

(a) -12

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

Calculus

Integrals

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

Integration Property [Swapping Limits]: $\displaystyle \int\limits^b_a {f(x)} \, dx = -\int\limits^a_b {f(x)} \, dx$

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

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

Integration Property [Splitting Integral]: $\displaystyle \int\limits^c_a {f(x)} \, dx = \int\limits^b_a {f(x)} \, dx + \int\limits^c_b {f(x)} \, dx$

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

Step-by-step explanation:

Step 1: Define

$\displaystyle \int\limits^6_4 {f(x)} \, dx = 5$

$\displaystyle \int\limits^4_{10} {f(x)} \, dx = 8$

$\displaystyle \int\limits^{10}_6 {[4f(x) + 10]} \, dx$

Step 2: Solve Pt. 1

[Integral] Rewrite [Integration Property - Addition]: $\displaystyle \int\limits^{10}_6 {[4f(x) + 10]} \, dx = \int\limits^{10}_6 {4f(x)} \, dx + \int\limits^{10}_6 {10} \, dx$ [Integral] Rewrite [Integration Property - Multiplied Constant]: $\displaystyle \int\limits^{10}_6 {[4f(x) + 10]} \, dx = 4\int\limits^{10}_6 {f(x)} \, dx + 10\int\limits^{10}_6 {} \, dx$

Step 3: Redefine

Manipulate the given integral values.

[Integrals] Combine [Integration Property - Splitting Integral]: $\displaystyle \int\limits^6_4 {f(x)} \, dx + \int\limits^4_{10} {f(x)} \, dx = \int\limits^6_{10} {f(x)} \, dx$ [Integral] Rewrite: $\displaystyle \int\limits^6_{10} {f(x)} \, dx = \int\limits^6_4 {f(x)} \, dx + \int\limits^4_{10} {f(x)} \, dx$ [Integral] Substitute in integrals: $\displaystyle \int\limits^6_{10} {f(x)} \, dx = 5 + 8$ [Integral] Add: $\displaystyle \int\limits^6_{10} {f(x)} \, dx = 13$ [Integral] Rewrite [Integration Property - Swapping Limits]: $\displaystyle -\int\limits^{10}_6 {f(x)} \, dx = 13$ [Integral] [Division Property of Equality] Isolate integral: $\displaystyle \int\limits^{10}_6 {f(x)} \, dx = -13$

Step 4: Solve Pt. 2

[Integral] Substitute in integral: $\displaystyle \int\limits^{10}_6 {[4f(x) + 10]} \, dx = 4(-13) + 10\int\limits^{10}_6 {} \, dx$ [Integral] Integrate [Integration Rule - Reverse Power Rule]: $\displaystyle \int\limits^{10}_6 {[4f(x) + 10]} \, dx = 4(-13) + 10(x) \bigg| \limits^{10}_6$ [Integral] Evaluate [Integration Rule - FTC 1]: $\displaystyle \int\limits^{10}_6 {[4f(x) + 10]} \, dx = 4(-13) + 10(10 - 6)$ [Integral] (Parenthesis) Subtract: $\displaystyle \int\limits^{10}_6 {[4f(x) + 10]} \, dx = 4(-13) + 10(4)$ [Integral] Multiply: $\displaystyle \int\limits^{10}_6 {[4f(x) + 10]} \, dx = -52 + 40$ [Integral] Add: $\displaystyle \int\limits^{10}_6 {[4f(x) + 10]} \, dx = -12$

Topic: AP Calculus AB/BC

Unit: Integration

Book: College Calculus 10e

Answer from: Quest

.there can be infinite values of m and n that can satisfy the above equation. for each value of m of m or n. there maybe infinite sets of solution for the above

step-by-step explanation:

Answer from: Quest

step-by-step explanation:

circumference divided by 3.141592654 then timesthe radius twice

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AP Calculus. The answer key says it's A) but I don't how to solve it.
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