- Using U-substitution to evaluate integrals is difficult to set up. It is sometimes hard to know which value will be the U and which value will be the DU.
- Finding the area between two curves is challenging when I forget to find which function is on top and when there are more than two places where the functions intersect.
- Finding the volume of a region using cylindrical shells is difficult because I often confuse this method with the washer method.
- Finding the volume of a region using the washer method is difficult because I often confuse this method with cylindrical shells.
- Using the limit definition to find the area under the curve also proves to be difficult because I do not remember the limit definition.
2. Do the Problems
- U-SUB
- Find the definite integral of x(1 - 3x2)4dx u=1-3x^2, du=-6xdx, -1/30(1-3x^2)^5 +c
- Find the definite integral of (1+sec(x))2(sec(x)tan(x))dx u= 1 + sec, du= sec(piX)tan(piX)dx, ((2-3x)^3)/3 +c
- Find f(x) when f(x) = x((2-3x)^1/2)dx u= 2-3x, du=-3dx, (2/45(2-3x)^5/2)(-4/27(2-3x)^3/2)+c
- Cylindrical Shells
- Find the volume of the region bounded by y = 2x3, y=2x, x = 1, and x = 2;
- revolved about the y- axis 2pi from 0 to 1 x(2x^3-2x)dx, 232/15pi
- Washer Method
- Find volume of area bounded by x = 0, y = x +4 and y = 12x2 revolved around x = -1 pi from -1 to 0, (x+4)^2 -(1/2x^2)^2dx, 737/60pi
- Limit Definition
- Evaluate the following using the limit definition and using integrationf(x) = 4x - 5 on [1, 5] The Summation of 4/n 4(1+4i/n) -5, The answer is 28.
- Area Between the Curves
- Find area of the region bounded by : f(x) = 2x-2 g(x) = -x2-2x+3 Interval -5 to 1, (-x^2-2x+3) - (2x-2), 108/3
- Find area of the region bounded by : f(x) = 2x3-1, g(x) = 2x-1 Interval 0 to 1, (2x-1)-(2x^3-1) The answer for this half is one half. We must multiply by to because the function is cubic and symmetrical across the y-axis. The answer is just one.
3. Create a problem that is similar.
- U-Sub
- Y= sec^2(x)tan^3(x), U= tan(x), Du= sec^2(x)dx, 1/4tan^4(x)+c
- Cylindrical Shells
- y=x-x^2, y=0, about x=2, 2pi integral of 0 to 1, (2-x)(x-x^2)dx, pi/2
- Washer Method
- y=x, y=x^2 about the x-axis. pi integral of 0 to 1, ((y^1/2)^2)-(y)^2dy, pi/6
- Limit Definition
- 2x^2-1 on [1, 5] limit as n approaches infinity of the summation of 4/n 2(1 + 4i/n)^2 -1, 236/3
- Area Between the Curves
- y= x^2-2x-4, y=x Integral -1 to 4, (x) - (x^2-2x-4)dx, 125/6
