Integrating Your Learning

Riemann Sums

100

Calculate the left Riemann sum for f(x) = x on the interval [0,2] using 2 subintervals.

Δx = (2−0)/2 = 1

Left Endpoints: x = 0, 1

 1(0) + 1(1) = 1

200

If the subinterval width (Δx) is 0.2 and there are 10 subintervals, what is the length of the interval [a,b]?

10 x 0.2 = 2

300

In a right Riemann sum, what happens to the approximation if the function f(x) increases on [a,b]?

The right Riemann sum overestimates the integral

400

Calculate the trapezoidal approximation for f(x)= x² on [1,9] with 4 subintervals.

Δx = (9-1)/4 =2

Endpoints: 1, 3, 5, 7, 9

2[(1+9)/2] + 2[(9+25)/2] + 2[(25+49)/2] + 2[(49+81)/2] = 248

Basic Integration

100

∫1/x dx = ?

ln|x| + C

200

∫₁² (3x²+1/x) dx = ?

7+ln2

300

∫_-17¹⁷ 17x¹⁷-1x¹+7x7 dx = ?

0

400

∫ (18x²+22)/(9+(3x)²) dx = ?

4/9 tan^-1(x) + 2x + C

U Substitution

100

∫₀³ 2x/(x²+1) dx = ?

ln10

200

∫ 2/(xlnx) dx = ?

2/ln(ln(x)) +C

300

∫ 3√(sec²x-1) dx = ?

-3ln(cosx) +C

400

∫ xe^{x^2} dx

1/2e^{x^2} + C

Application Problems

100

A particle moves along a straight line with a velocity v(t) = 3t m/s. What is the total displacement of the particle over the interval [0,2]?

∫₀² 3tdt = [3t²/2]₀² = 3(2)²/2 - 3(0)²/2

= 6m

200

Find the area under the curve f(x)= x² on the interval [1,3].

∫₁³ x²dx = [x³/3]₁³ = 27/3 - 1/3 = 26/3

300

A particle’s velocity is given by v(t) = 3t² - 2t + 1 m/s. Find the particle’s total displacement over the interval [0,3].

∫₀³ (3t² - 2t + 1)dt = [t³ - t² + t]₀³ = 27 - 9 + 3 = 21 m

400

Water flows into a tank at a rate given by r(t)=4t² liters per minute. How much water flows into the tank over the first 5 minutes?

∫₀⁵ 4t²dx = 4[t³/3]₀⁵ = 4 x 125/3 = 500/3 liters

Funfunfun

100

sin²(x)+cos²(x) = ?

1

200

which function stays the same whether you take the derivative or the integral?

e^x

300

When doing indefinite integrals, what will you always remember? 

+C

400

What is the derivative of inverse secant?

1  /  |x| √(x²-1)