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Definite Integrals

100

∫₀³(3x - 6)dx

What is -4.5?

200

∫₀²(6x² + 9x + 3)dx

What is 40?

300

The length of the curve y=6x^3 from x=2 to x=5.

What is 702.008?

400

∫₀^♾️(x/(x² + 3))dx

The integral is divergent.

500

∫₀⁶6/(x²-1)dx

What is -1.009?

Indefinite Integrals

100

∫sinxcosxdx

What is (sin²x)/2 + c?

200

∫(3x² + 16x²¹ + 7)dx

What is x³ + (8x/11)x²² + 7x + c?

300

∫(e^(2x))/(1 + e^(4x))dx

What is (1/2)arctan(e^(2x)) + c?

400

∫(e^x^1/3)/(x^2/3)dx

What is 3e^x^1/3 + c?

500

∫(-x)/((1-x⁴)^1/2)dx

What is (1/2)arcsin(x²) + c?

Series

100

Does Σ1/n⁴ (n=1 to n=♾️) converge or diverge?

The series converges by the p-series test (p=4>1).

200

Does Σ(pi/e)ⁿ (from n=1 to n=♾️) converge or diverge?

The series diverges by the geometric series test (r=pi/e, r>1).

300

Does Σ(n²+4)/(3n³-2n+4) (from n=1 to ♾️)  converge or diverge?

The series diverges by the Limit Comparison Test (compared to the harmonic series).

400

Does Σ(n+1)!/5ⁿ from n=1 to ♾️ converge or diverge?

The series diverges by the Ratio Test.

500

Find the third-degree Taylor polynomial approximation for f(x)=3x⁴+43x+2 centered around x=2.

P₃(x)=136 + 139(x-2) + (144/2!)(x-2)² + (144/3!)(x-2)³

Area and Volume

100

Area of the region between y=e^3x and the x-axis from x=0 to x=3.

What is ∫₀³(e^3x)dx = 2700.694.

200

Area of the region between y=x²+3 and y=2x+3.

What is ∫₀²((2x-3)-(x²+3))dx = 1.333

300

Volume of the solid formed when the region between y=2x^1/2 and the x-axis from x=0 to x=3 is rotated around the x-axis.

What is pi∫₀³(2x^1/2)²dx = 18pi

400

Volume of the solid formed when the region between y=2x² and y=8x is rotated around y=-2. 

What is pi∫₀⁴((8x+2)²-(2x²+2)²)dx = 1983.811?

500

A solid has a base that can be represented by the region bounded by y=(1/2)x² and the x-axis from x=0 to x=4. The cross-sections are semicircles. Find the volume of the solid.

What is (pi/32)∫₀⁴(x⁴)dx = 20.106?

Limits and Continuity

100

lim_x->0(7x+28)/(x²+x-12) 

What is -7/3?

200

lim_x->♾️(x²-4x)/(4x²+3)

What is 1/4?

300

lim_x->2(x²-x-2)/(x²-2x)

What is 1/2?

400

h(x) = x² , x≤5

          x+k, x>5
Find the value of k that makes h(x) continuous at x=5.

What is 20?

500

f(x)= x+1 , x<1

        ax+b , 1≤x<2
        3x     , x≥2

Find the values of a and b that make f(x) continuous at x=1 and x=2.

What is a=4 and b=-2?