section 5.2b. do now: exploration 1 on page 264 it is a fact that with this information, determine...
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INTEGRALS ON THE CALCULATOR, DISCONTINUOUS INTEGRABLE FUNCTIONS
Section 5.2b
Do Now: Exploration 1 on page 264
It is a fact that 0sin 2x dx
With this information, determine the values of the followingintegrals. Explain your answers (use a graph, when necessary).
2sin x dx
21.
2
0sin x dx
02.
2
0sin x dx
13.
02 sin x dx
2 2 4.
02sin x dx
45.
2
2sin 2x dx
26.
Do Now: Exploration 1 on page 264
It is a fact that 0sin 2x dx
With this information, determine the values of the followingintegrals. Explain your answers (use a graph, when necessary).
sinu du
07.
2
0sin 2x dx
48.
0cos x dx
09.
sink
kx dx
010. Suppose k is any positive number. Make a conjectureabout
A Similar Challenge: #29-38 on p.267-268Use graphs, your knowledge of area, and the fact that
1 3
0
1
4x dx to evaluate the given integrals.
1 3
1x dx
29. 0
1 3
03x dx30.
13
4
3 3
22x dx31.
1
4
1 3
1x dx
32.1
2
1 3
01 x dx33.
3
4
2 3
11x dx
34.
1
4
A Similar Challenge: #29-38 on p.267-268Use graphs, your knowledge of area, and the fact that
1 3
0
1
4x dx to evaluate the given integrals.
32
0 2
xdx
35.
1
2
8 3
8x dx
36. 0
1 3
01x dx37.
3
4
13
0xdx38.
3
4
Integrals on the CalculatorOur modern calculators are good at calculating Riemannsums…our text denotes this function as NINT:
b
af x dx NINT , , ,f x x a b
We write this statement with an understanding that the right-hand side of the equation is an approximation of the left-handside…
Integrals on the CalculatorExamples: Evaluate the following integrals numerically.
2
1sinx x dx
2.043
1
20
4
1dx
x 3.142
25
0
xe dx 0.886
Discontinuous Integrable FunctionsAs we already know, a function is not differentiable where it isdiscontinuous. However, we can integrate functions that havepoints of discontinuity. Examples…
–1
Find2
1
xdx
xLet’s look at the graph…
1 2–1
1
Areas of rectangles:
2
11 2 1
xdx
x
Discontinuityat x = 0!!!
What does our calculatorgive us on this one???
Discontinuous Integrable FunctionsAs we already know, a function is not differentiable where it isdiscontinuous. However, we can integrate functions that havepoints of discontinuity. Examples…
Explain why the given function is not continuous on [0, 3].What kind of discontinuity occurs?
2 4
2
xf x
x
Removable discontinuity at x = 2
Discontinuous Integrable FunctionsAs we already know, a function is not differentiable where it isdiscontinuous. However, we can integrate functions that havepoints of discontinuity. Examples…
Use areas to show that 3 2
0
410.5
2
xdx
x
The thin strip above x = 2 haszero area, so the area under thecurve is the same as
3
02x dx 13 2 5 10.5
2
A Trapezoid!!!
Discontinuous Integrable FunctionsAs we already know, a function is not differentiable where it isdiscontinuous. However, we can integrate functions that havepoints of discontinuity. Examples…
Use areas to show that 5
0int 10x dx
Sum the rectangles:
5
0int x dx
1 1 2 3 4 10