Question

Which definite integral would you use to compute the area enclosed by the parabola y=-x^2+4 and the line y=2x-3?

Answer 1

`\displaystyle A = \int\limits^(2√(2) - 1)_(-(2√(2) + 1)) {(-x^2 - 2x + 7)} \, dx`

General Formulas and Concepts:

Calculus

Integration

• Integrals

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

Explanation:

Step 1: Define

Identify

y = -x² + 4

y = 2x - 3

Step 2: Identify

Graph functions and find region and bounds of integration.

Bounds: [-(2√2 + 1), 2√2 - 1]

Step 3: Find Area

1. Substitute in variables [Area of a Region Formula]:
   `\displaystyle A = \int\limits^(2√(2) - 1)_(-(2√(2) + 1)) {[-x^2 + 4 - (2x - 3)]} \, dx`
2. Simplify:
   `\displaystyle A = \int\limits^(2√(2) - 1)_(-(2√(2) + 1)) {(-x^2 - 2x + 7)} \, dx`

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

Unit: Integration