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Sagot :
Answer:
[tex]\displaystyle \int\limits^{1.718}_{-1.529} {\sqrt{7 + x} - x^2} \, dx = 5.74773[/tex]
General Formulas and Concepts:
Calculus
Differentiation
- Derivatives
- Derivative Notation
Derivative Property [Addition/Subtraction]: [tex]\displaystyle \frac{d}{dx}[f(x) + g(x)] = \frac{d}{dx}[f(x)] + \frac{d}{dx}[g(x)][/tex]
Basic Power Rule:
- f(x) = cxⁿ
- f’(x) = c·nxⁿ⁻¹
Integration
- Integrals
Integration Rule [Reverse Power Rule]: [tex]\displaystyle \int {x^n} \, dx = \frac{x^{n + 1}}{n + 1} + C[/tex]
Integration Rule [Fundamental Theorem of Calculus 1]: [tex]\displaystyle \int\limits^b_a {f(x)} \, dx = F(b) - F(a)[/tex]
Integration Property [Addition/Subtraction]: [tex]\displaystyle \int {[f(x) \pm g(x)]} \, dx = \int {f(x)} \, dx \pm \int {g(x)} \, dx[/tex]
U-Substitution
Area of a Region Formula: [tex]\displaystyle A = \int\limits^b_a {[f(x) - g(x)]} \, dx[/tex]
Step-by-step explanation:
Step 1: Define
[tex]\displaystyle \left \{ {{y = x^2} \atop {y = \sqrt{7 + x}}} \right.[/tex]
Step 2: Identify
Graph the systems of equations - see attachment.
Top Function: [tex]\displaystyle y = \sqrt{7 + x}[/tex]
Bottom Function: [tex]\displaystyle y = x^2[/tex]
Bounds of Integration: [-1.529, 1.718]
Step 3: Integrate Pt. 1
- Substitute in variables [Area of a Region Formula]: �� [tex]\displaystyle \int\limits^{1.718}_{-1.529} {\sqrt{7 + x} - x^2} \, dx[/tex]
- [Integral] Rewrite [Integration Property - Addition/Subtraction]: [tex]\displaystyle \int\limits^{1.718}_{-1.529} {\sqrt{7 + x} - x^2} \, dx= \int\limits^{1.718}_{-1.529} {\sqrt{7 + x}} \, dx - \int\limits^{1.718}_{-1.529} {x^2} \, dx[/tex]
- [Right Integral] Integration Rule [Reverse Power Rule]: [tex]\displaystyle \int\limits^{1.718}_{-1.529} {\sqrt{7 + x} - x^2} \, dx= \int\limits^{1.718}_{-1.529} {\sqrt{7 + x}} \, dx - \frac{x^3}{3} \bigg| \limits^{1.718}_{-1.529}[/tex]
- Evaluate [Integration Rule - Fundamental Theorem of Calculus 1]: [tex]\displaystyle \int\limits^{1.718}_{-1.529} {\sqrt{7 + x} - x^2} \, dx= \int\limits^{1.718}_{-1.529} {\sqrt{7 + x}} \, dx - 2.88176[/tex]
Step 4: Integrate Pt. 2
Identify variables for u-substitution.
- Set u: [tex]\displaystyle u = 7 + x[/tex]
- [u] Basic Power Rule [Derivative Rule - Addition/Subtraction]: [tex]\displaystyle du = dx[/tex]
- [Limits] Switch: �� [tex]\displaystyle \left \{ {{x = 1.718 ,\ u = 7 + 1.718 = 8.718} \atop {x = -1.529 ,\ u = 7 - 1.529 = 5.471}} \right.[/tex]
Step 5: Integrate Pt. 3
- [Integral] U-Substitution: [tex]\displaystyle \int\limits^{1.718}_{-1.529} {\sqrt{7 + x} - x^2} \, dx= \int\limits^{8.718}_{5.471} {\sqrt{u}} \, du - 2.88176[/tex]
- [Integral] Integration Rule [Reverse Power Rule]: [tex]\displaystyle \int\limits^{1.718}_{-1.529} {\sqrt{7 + x} - x^2} \, dx = \frac{2x^\Big{\frac{3}{2}}}{3} \bigg| \limits^{8.718}_{5.471} - 2.88176[/tex]
- Evaluate [Integration Rule - Fundamental Theorem of Calculus 1]: [tex]\displaystyle \int\limits^{1.718}_{-1.529} {\sqrt{7 + x} - x^2} \, dx = 8.62949 - 2.88176[/tex]
- Simplify: [tex]\displaystyle \int\limits^{1.718}_{-1.529} {\sqrt{7 + x} - x^2} \, dx = 5.74773[/tex]
Topic: AP Calculus AB/BC (Calculus I/I + II)
Unit: Integration
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