8. Definite Integrals

Properties of integrals Use only the fact that ∫₀⁴ 3𝓍 (4 ―𝓍) d𝓍 = 32, and the definitions and properties of integrals, to evaluate the following integrals, if possible.

(c) ∫₄⁰ 6𝓍(4 ― 𝓍) d(𝓍)

Using properties of integrals Use the value of the first integral I to evaluate the two given integrals. 

I = ∫₀¹ (𝓍³ ― 2𝓍) d𝓍 = ―3/4

(a) ∫₀¹ (4𝓍―2𝓍³) d𝓍

Evaluating integrals Evaluate the following integrals.

∫₋₂² (3𝓍⁴―2𝓍 + 1) d𝓍

75. Exploring powers of sine and cosine

e. Repeat parts (a), (b), and (c) with sin²x replaced by sin⁴x. Comment on your observations.

Lifetime of a tire Assume the random variable L in Example 2f is normally distributed with mean μ = 22,000 miles and σ = 4,000 miles.

a. In a batch of 4000 tires, how many can be expected to last for at least 18,000 miles?

Symmetry in integrals Use symmetry to evaluate the following integrals.

∫²₋₂ [(x³ ― 4x) / (x² + 1)] dx 

7–64. Integration review Evaluate the following integrals.

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

Evaluate the line integral of the function $f(x,y)=xy$ along the curve $C$, where $C$ is given by the parametric equations $x=t^2$, $y=2t$ for $0\le t\le 5$. That is, compute ${\int }_C^{}xyds$.

Definite integrals Use a change of variables or Table 5.6 to evaluate the following definite integrals.                                                                                                                         

 ∫₀¹ 2e²ˣ d𝓍

Evaluate the iterated integral: ${\int }_0^2{\int }_0^1{\left(x+y\right)}^{2}dxdy$

Evaluate the definite integral: ${\int }_1^4(t^3-t^2-4) dt$.

Find the area of the part of the plane $5x+4y+z=20$ that lies in the first octant.

Consider the double integral ${\int }_0^2{\int }_{y^2}^{4}f(x,y)dxdy$. Which of the following correctly expresses the integral with the order of integration reversed?

Suppose the graph of $f$ consists of two regions between $x=0$ and $x=4$: from $x=0$ to $x=2$, $f\left(x\right)$ forms a triangle above the $x$-axis with area $3$; from $x=2$ to $x=4$, $f\left(x\right)$ forms a rectangle below the $x$-axis with area $4$. What is the value of the definite integral ${\int }_0^{4}f\left(x\right)dx$?

Evaluate the definite integral: ${\int }_{-2}^3(x^2-3) dx$.