Question

Area of the bounded curves y=x^2, y=√(7+x)

Answer 1

`\displaystyle \int\limits^(1.718)_(-1.529) {√(7 + x) - x^2} \, dx = 5.74773`

General Formulas and Concepts:

Calculus

Differentiation

• Derivatives
• Derivative Notation

Derivative Property [Addition/Subtraction]:
`\displaystyle (d)/(dx)[f(x) + g(x)] = (d)/(dx)[f(x)] + (d)/(dx)[g(x)]`

Basic Power Rule:

1. f(x) = cxⁿ
2. f’(x) = c·nxⁿ⁻¹

Integration

• Integrals

Integration Rule [Reverse Power Rule]:
`\displaystyle \int {x^n} \, dx = (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 [Addition/Subtraction]:
`\displaystyle \int {[f(x) \pm g(x)]} \, dx = \int {f(x)} \, dx \pm \int {g(x)} \, dx`

U-Substitution

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

Explanation:

Step 1: Define

`\displaystyle \left \{ {{y = x^2} \atop {y = √(7 + x)}} \right.`

Step 2: Identify

Graph the systems of equations - see attachment.

Top Function:
`\displaystyle y = √(7 + x)`

Bottom Function:
`\displaystyle y = x^2`

Bounds of Integration: [-1.529, 1.718]

Step 3: Integrate Pt. 1

1. Substitute in variables [Area of a Region Formula]:
   `\displaystyle \int\limits^(1.718)_(-1.529) {√(7 + x) - x^2} \, dx`
2. [Integral] Rewrite [Integration Property - Addition/Subtraction]:
   `\displaystyle \int\limits^(1.718)_(-1.529) {√(7 + x) - x^2} \, dx= \int\limits^(1.718)_(-1.529) {√(7 + x)} \, dx - \int\limits^(1.718)_(-1.529) {x^2} \, dx`
3. [Right Integral] Integration Rule [Reverse Power Rule]:
   `\displaystyle \int\limits^(1.718)_(-1.529) {√(7 + x) - x^2} \, dx= \int\limits^(1.718)_(-1.529) {√(7 + x)} \, dx - (x^3)/(3) \bigg| \limits^(1.718)_(-1.529)`
4. Evaluate [Integration Rule - Fundamental Theorem of Calculus 1]:
   `\displaystyle \int\limits^(1.718)_(-1.529) {√(7 + x) - x^2} \, dx= \int\limits^(1.718)_(-1.529) {√(7 + x)} \, dx - 2.88176`

Step 4: Integrate Pt. 2

Identify variables for u-substitution.

1. Set u:
   `\displaystyle u = 7 + x`
2. [u] Basic Power Rule [Derivative Rule - Addition/Subtraction]:
   `\displaystyle du = dx`
3. [Limits] Switch:
   `\displaystyle \left \{ {{x = 1.718 ,\ u = 7 + 1.718 = 8.718} \atop {x = -1.529 ,\ u = 7 - 1.529 = 5.471}} \right.`

Step 5: Integrate Pt. 3

1. [Integral] U-Substitution:
   `\displaystyle \int\limits^(1.718)_(-1.529) {√(7 + x) - x^2} \, dx= \int\limits^(8.718)_(5.471) {√(u)} \, du - 2.88176`
2. [Integral] Integration Rule [Reverse Power Rule]:
   `\displaystyle \int\limits^(1.718)_(-1.529) {√(7 + x) - x^2} \, dx = \frac{2x^\Big{(3)/(2)}}{3} \bigg| \limits^(8.718)_(5.471) - 2.88176`
3. Evaluate [Integration Rule - Fundamental Theorem of Calculus 1]:
   `\displaystyle \int\limits^(1.718)_(-1.529) {√(7 + x) - x^2} \, dx = 8.62949 - 2.88176`
4. Simplify:
   `\displaystyle \int\limits^(1.718)_(-1.529) {√(7 + x) - x^2} \, dx = 5.74773`

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

Unit: Integration