Finding volume

The region in the first quadrant enclosed by the coordinate axes, the curve y = e^x, and the line x = 1 is revolved about the y-axis to generate a solid. Find the volume of the solid.

Use the substitutions in Equations (1)–(4) to evaluate the integrals in Exercises 33–40. Integrals like these arise in calculating the average angular velocity of the output shaft of a universal joint when the input and output shafts are not aligned.

∫ dx / (1 + sin x + cos x)

Finding volume

The region in the first quadrant that is enclosed by the x-axis and the curve y = 3x√(1 − x) is revolved about the y-axis to generate a solid. Find the volume of the solid.

Use the substitutions in Equations (1)–(4) to evaluate the integrals in Exercises 33–40. Integrals like these arise in calculating the average angular velocity of the output shaft of a universal joint when the input and output shafts are not aligned.

∫(from π/2 to 2π/3) cos θ dθ / (sin θ cos θ + sin θ)

Evaluate the limits in Exercise 9 and 10 by identifying them with definite integrals and evaluating the integrals.

lim (n → ∞) Σ (from k=1 to n) ln √(1 + k/n)

Use the substitution z = tan(θ/2) to evaluate the integrals in Exercises 41 and 42.

∫ csc θ dθ

Finding arc length

Find the length of the curve

y = ∫ from 0 to x of √(cos(2t)) dt, 0 ≤ x ≤ π/4.

Evaluate the integrals in Exercises 1–6.

∫ dt / (t - √(1 - t²))

2. When applying the formula for integration by parts, how do you choose the u and dv? How can you apply integration by parts to an integral of the form ∫ f(x) dx?

Evaluate the integrals in Exercises 37–44.

∫ sec²(θ) sin³(θ) dθ

You are planning to use Simpson’s Rule to estimate the value of the integral Estimate ∫ from 1 to 2 of f(x) dx with an error magnitude less than 10⁻⁵ using Simpson’s Rule.

You have determined that |f⁽⁴⁾(x)| ≤ 3 throughout the interval of integration. How many subintervals should you use to ensure the required accuracy?

(Remember that for Simpson’s Rule the number of subintervals must be even.)

Evaluate the integrals in Exercises 69–134. The integrals are listed in random order so you need to decide which integration technique to use.

∫ (z + 1) / [z²(z² + 4)] dz

Evaluate the integrals in Exercises 69–134. The integrals are listed in random order so you need to decide which integration technique to use.

∫ (2 − cosx + sinx) / sin²x dx

Evaluate the integrals in Exercises 69–134. The integrals are listed in random order so you need to decide which integration technique to use.

∫ t dt / √(9 − 4t²)

Which of the improper integrals in Exercises 63–68 converge and which diverge?

∫ from −∞ to ∞ of (2 / (e^x + e^(−x))) dx

Evaluate the improper integrals in Exercises 53–62.

∫ from −∞ to ∞ of (1 / (4x² + 9)) dx

Evaluate the integrals in Exercises 69–134. The integrals are listed in random order so you need to decide which integration technique to use.

∫ e^t dt / (e^(2t) + 3e^t + 2)

Evaluate the integrals in Exercises 69–134. The integrals are listed in random order so you need to decide which integration technique to use.

∫₋₁⁰ e^x / (e^x + e^(−x)) dx

Evaluate the integrals in Exercises 9–28. It may be necessary to use a substitution first.

∫ [x / (x² + 4x + 3)] dx

Evaluate the integrals in Exercises 69–134. The integrals are listed in random order so you need to decide which integration technique to use.

∫ (sin5t) dt / [1 + (cos5t)²]

Evaluate the integrals in Exercises 9–28. It may be necessary to use a substitution first.

∫ [cos(θ) / (sin²(θ) + sin(θ) − 6)] dθ

Evaluate the improper integrals in Exercises 53–62.

∫ from 0 to ∞ of (x² * e^(−x)) dx

Evaluate the integrals in Exercises 69–134. The integrals are listed in random order so you need to decide which integration technique to use.

125. ∫ dx / (√x * √(1 + x))

Evaluate the integrals in Exercises 1–8 using integration by parts.

∫ x² sin(1 − x) dx

Evaluate the improper integrals in Exercises 53–62.

∫ from 3 to ∞ of (2 / (u² − 2u)) du

Evaluate the integrals in Exercises 69–134. The integrals are listed in random order so you need to decide which integration technique to use.

∫ √(2x − x²) dx

Evaluate the integrals in Exercises 9–28. It may be necessary to use a substitution first.

∫ [1 / √(e^s + 1)] ds

Evaluate the integrals in Exercises 69–134. The integrals are listed in random order so you need to decide which integration technique to use.

∫ 9 dv / (81 − v⁴)

Evaluate the integrals in Exercises 29–32 (b) using a trigonometric substitution.

∫ [y / √(16 − y²)] dy

Heat capacity of a gas

Heat capacity 

C_v

is the amount of heat required to raise the temperature of a given mass of gas with constant volume by 1°C, measured in units of cal/deg-mol (calories per degree gram molecular weight).

The heat capacity of oxygen depends on its temperature T and satisfies the formula

C_v = 8.27 + 10^(-5) * (26T − 1.87T²)

Use Simpson’s Rule to find the average value of C_v and the temperature at which it is attained for

20°C ≤ T ≤ 675°C.