Integration
The definite integral and the primitive function
1. Evaluate 23
0 , 2
0
3 and show that 2
3
0 = 2
0
3
2. Evaluate 22
1 , 2
3
2 , 2
3
1 and show that 2
3
1 = 2
2
1 + 2
3
2
3. Evaluate 23
1 , 2
3
1 , (2 + 2)
3
1 and verify that
(2 + 2)3
1 = 2
3
1 + 2
3
1
4. Evaluate (42 + 8)3
0 , 4 (2 + 2)
3
0 , and verify that
(42 + 8)3
0 = 4 (2 + 2)
3
0
*Note: These are important general rules for integration (or
tricks if you will)
5. Evaluate:
a. (2 5)1
0
b. (2 2)3
2
6. Find the area of the region bounded by the x axis and the
graph y = 2 x x2.
7. Find the area of the region bounded by the x axis and the
graph of y = x2(1-x).
More areas
1. Calculate the area of the region bounded by the curve y = 4
x2 and the x axis.
2. Calculate the area of the region bounded by the curve y x3,
the x axis and the ordinates
x=-3 and x=3.
3. Find a positive number k such that the area of the region
bounded by the graph of
f(x) = kx(2-x)2 and the x axis is equal to 1 square unit.
4. Calculate the area of the region bounded by the curve y =
(x+1)(x-1)(x-3), the x axis and the
ordinates at x=0 and x=2.
Trapezoidal Rule
1. Evaluate 41
1 using the trapezoidal rule with 4 subintervals.
2. Evaluate 1 + 32
0 using the trapezoidal rule with two subintervals.
Simpsons Rule
1. Evaluate 41
1 using the Simpsons Rule with five function values.
2. Use Simpsons rule with 9 function values to evaluate
1+
4
0 correct to 5 decimal places.
Area between two curves
1. Calculate the area of the region enclosed by the graphs of
the parabola y = x2 + 4 and the
line y = 5.
2. Calculate the area of the region enclosed by the graphs of
the parabolas f(x) = 6x2 5x and
g(x) = 5x 4x2.
3. Calculate the area bounded by the inequalities yx and y3x
x2.
Area bounded by the y-axis
1. Calculate the area of the region bounded by the curve =1
, the y axis and the lines y=1 and
y=4 using the trapezoidal rule with 3 subintervals.
2. Calculate the area of the region bounded by the lines y = 4
x, y = x and the y axis.
Volume of solids of revolution
1. Find the volume of the solid formed by rotating about the x
axis the arc of the parabola y=x2
between x=0 and x=3.
2. The semicircle = 9 2 is rotated about the x axis. Calculate
the volume of the sphere
generated.
3. Find the volume of the solid formed when the region bounded
by the parabola y=4-x2 and
the line y=1 is rotated about the y-axis.
