Reference no: EM133541500

Integrals

Problem # 1

Use the definition of improper integral to evaluate the following integral (be attentive with the signs, use limit for explanation)

∫_-∞⁰ x·e^-2x2 dx =

Problem # 2

Evaluate (clearly state substitution and keep the exact answer with radical signs)

∫₀² x² √(x³+4 ) dx =

Problem # 3

Use integration by parts to evaluate.

Clearly state u=..,du=...,v=...,dv=...,

∫x²·ln?(2x)dx?

Problem # 4

Use partial fraction decomposition to evaluate and show the procedure of finding coefficients

∫ 7x/(x² + 3x - 70) dx

Problem # 5

Use Trigonometric Substitution(fully complete the problem by applying "right triangle" if applicable)

∫x²/√(4-x² ) dx

Problem # 6

Evaluate the following Trigonometric integral

∫₀^π/3 cos³?(2x)dx

Sequences and Series

Problem # 7

Find limit of the terms of the sequence {(1+2/n)⁵ⁿ }. Do not just use formula, apply LH' Rule

lim_n→∞? (1+2/n)⁵ⁿ

State whether sequence converges or diverges.

Problem # 8

Use the Limit Comparison Test to determine whether the given series converge or diverge:

clearly state b_n

justify why ∑b_n diverges or converges

find the lim_n→∞?|a_n/b_n

clearly state your conclusion

∑_n=10^∞ (√n-5)/(n² √n+n)

Problem # 9

Determine whether the given series Converge Absolutely, Converge Conditionally, or Diverge. Give reasons for your conclusion: state the test(s) name(s) and clearly show the procedure.

∑_n=3^∞ (-1)ⁿ⁺¹ 5/√n

Problem 10

(a) find all values of x for which each given power series converges
(b) check both end points (if any)
(c) state the interval of convergence for the series in interval notation

∑_n=1^∞ (x+2)ⁿ/(n² 2ⁿ)

Problem # 11
Calculate the Taylor polynomial P₂ for the given function centered at the given value of C (state three nonzero terms). Be attentive that c=9

f(x) = 1/√x, c = 9

Application problems

Problem # 12

The velocity of a car after t seconds is (30-5t) feet per second.
(a) How many seconds does it take for the car to come to a stop (v= 0)?
(b) How far does the car travel while coming to a stop?

Problem # 13
Find area between two curves y₁ = 4x +16 and y₂ = 2x² + 10

Problem # 14
Let R be the region bounded by the following curves (in the 1st Quadrant)

y=6√x,   y=2x

R is revolved about the x-axis.

Set upand evaluate an integral that gives the Volume of the solid.