8. Definite Integrals

Given the region bounded by $y=x^2$, $x=0$, and $y=4$, determine by direct integration the x-coordinate of the centroid of the area.

Find the area of the region enclosed by one loop of the curve $r=\sin \left(8\theta \right)$.

Find ${\int }_0^{6}f\left(x\right)dx$ given the graph of $y=f\left(x\right)$.

Given the following definite integral of the function $f\left(x\right)=3x^2-2x$, write the simplified integral:

$-\int_4^0\!f\left(x\right)\,dx$

Write the two definite integrals subtracted below as a single integral.

${\int }_1^6\sqrt{x^2-5x}dx-{\int }_{10}^6\sqrt{x^2-5x}dx$

Evaluate the following definite integral.

${\int }_{100}^{100}\frac{\sin \left(2x\right)-100x^2}{\sqrt{x^5+\tan (3x-6)}}dx$

7–64. Integration review Evaluate the following integrals.

12. ∫ from -5 to 0 of dx / √(4 - x)

7–64. Integration review Evaluate the following integrals.

24. ∫ from 0 to θ of (x⁵⸍² - x¹⸍²) / x³⸍² dx

7–84. Evaluate the following integrals.

19. ∫ from 0 to π/2 [sin⁷x] dx

7–64. Integration review Evaluate the following integrals.

60. ∫ from 0 to π/4 of 3√(1 + sin 2x) dx

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

(a) If ƒ is symmetric about the line 𝓍 = 2 , then ∫₀⁴ ƒ(𝓍) d𝓍 = 2 ∫₀² ƒ(𝓍) d𝓍.

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. Assume ƒ and ƒ' are continuous functions for all real numbers.

(a) A(𝓍) = ∫ₐˣ ƒ(t) dt and ƒ(t) = 2t―3 , then A is a quadratic function.

Prove the following orthogonality relations (which are used to generate Fourier series). Assume m and n are integers with m ≠ n.

c.

π

∫ sin(mx) cos(nx) dx = 0, when |m + n| is even

0

49–63. {Use of Tech} Integrating with a CAS Use a computer algebra system to evaluate the following integrals. Find both an exact result and an approximate result for each definite integral. Assume a is a positive real number.

52. ∫ from 0 to π/2 of cos⁶x dx

49–63. {Use of Tech} Integrating with a CAS Use a computer algebra system to evaluate the following integrals. Find both an exact result and an approximate result for each definite integral. Assume a is a positive real number.

58. ∫₀^{2π} dt / (4 + 2 sin t)²