Question

Which of the following integrals represents the area of the region bounded by x = 0, x = 1 and the functions f(x) = -2x4 and g(x) = x2?

the integral from 0 to 1 of the quantity, x squared plus 2 times x to the 4th power, dx

the integral from 0 to 1 of the quantity, x squared minus 2 times x to the 4th power, dx

the integral from 0 to 1 of the quantity, negative 2 times x to the 4th power minus x squared, dx

the integral from 0 to 1 of the quantity, x to the 4th power minus 4 times x to the 8th power, dx

Answer 1

A.
`\displaystyle \int\limits^1_0 {(x^2 + 2x^4)} \, dx`

General Formulas and Concepts:

Algebra I

Functions

• Function Notation
• Graphing

Calculus

Integration

• Integrals
• Definite Integrals
• Integration Constant C
• Area under the curve/area between 2 curves

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

• f(x) is top function, g(x) is bottom function

Explanation:

Step 1: Define

Identify

f(x) = -2x⁴

g(x) = x²

Interval bound [0, 1]

Step 2: Visualize

Graph the given and identify more information. See attachment.

Top function: g(x)

Bottom function: f(x)

Bounds of Integration: [0, 1]

Step 3: Find Area

1. Substitute in variables [Area of a Region Formula]:
   `\displaystyle A = \int\limits^1_0 {[x^2 - (-2x^4)]} \, dx`
2. [Integral] Simplify:
   `\displaystyle A = \int\limits^1_0 {(x^2 + 2x^4)} \, dx`

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

Unit: Integration

Book: College Calculus 10e