intigration
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math260_w7_lab_worksheet.docxName_____________________
Category 1: Area Under the Curve Part II
1. What is the area under the curve in the interval [-3, 3] for the function ?
What happens when you find the value of the definite integral ? Why can’t this be the area under the curve above? How would you go about determining the area under the curve so that you avoid the cancelling out nature of the area above the x-axis with the area below the x-axis?
The value of function f(x) is psitive for x>0 and negative for x<0
so if you intergrate ??x^3-9x^2 dx?from -3 to +3, the negative integral value from (-3,0) will cancel out the positive integral value from (0,3) and resulting value will be 0 if we have to find the area of the f(x) in term of magnitude then we have to calculate the below integrals
?_(-3)^0?|x^3-9x^2| dx?+ ?_(0)^3?|x^3-9x^2| dx?
so Area under the curve is 121.5 unit^2
the formula for the integral would be ?_(-3)^3 |?x^3-9x^2 ?| dx and not ?_(-3)^3?x^3-9x^2 dx? .
The reason for this is that we always assume area to be a positive quantity while integral can be positive or -ve.
So when integral is +ve then it is equal to area but when it is -ve then its absolute value is equal to area.
So the problem can be solved if we integrate it in regions i.e. we break integrals from -3 to 0 and 0 to 3.
for 0 to 3 regions we can take the - times function as function is -ve in that interval,
In general for avoiding this error we first find out zeros and then see the regions in which the function is +ve or -ve by the use of derivatives.
2. Use your graphing calculator to graph this function . Make sure your window is set to x-min = -4, x – max = 5, y – min = -75, y-max = 50. Draw a sketch of the graph.
3. Evaluate the definite integral below by hand using the Fundamental Theorem of calculus
. What value do you get? Can this be the actual area under the curve?
Why or Why Not?
4. How can we break the area under the curve we are finding into smaller pieces so that we can find the true value?
5. Solve the following definite integrals and use them to find the area under the curve.
6. Find the area under the curve for .
Category 2: Substitution
1. Integral:
Let u = , then du = 2x dx
Substitution:
Solution: , thus
2. Integral: dx
Let u =, then du = 6x dx
Substitution:
Solution:, thus
3. Integral:
Let u = , then du =
Substitution:
Solution: , thus + C
How is u determined? How is du determined? How does knowing these two items allow us to make the subsequent substitution? Why did 1/6 and 1/20 appear outside the integration symbol in examples 2 and 3? What condition is necessary to make substitution work? What are the steps in the process of using substitution to integrate?
Solve these integration problems
1. 3.
2. 4.
Category 3: Logarithmic Form
1. Integral:
Solution:
2. Integral:
Solution:
3. Integral:
Solution:
4. Integral:
Solution:
What is the formula for ? How is substitution being used to solve examples 2 and 3?
Why can’t we solve using the formula above?
Solve these integrals
1. 3.
2. 4.
Category 4: Exponential Integration
1. Integral:
Solution:
2. Integral:
Solution:
3. Integral:
Solution:
4. Integral:
Solution:
What is the formula for ? How is substitution being used to help solve examples 2, 3, and 4?
Solve these integrals
1. 3.
2. 4.
Category 5: Sine and Cosine Integration
1. Integral:
Solution:
2. Integral:
Solution:
3. Integral:
Solution:
4. Integral:
Solution: + C
What are the formulas for ?
Solve these integrals
1. 2.
Category 6: Tangent and Cotangent Integration
1. Integral:
Solution:
2. Integral:
Solution:
3. Integral:
Solution:
4. Integral:
Solution:
What are the formulas for ?
Solve these integrals:
1. 2.
Category 7: Secant and Cosecant integration
1. Integral:
Solution:
2. Integral:
Solution:
3. Integral:
Solution:
4. Integral:
Solution:
What are the formulas for ?
Solve these integrals
1. 2.
Category 8: Other Trig Formulas
1. Integral:
Answer:
2. Integral:
Solution:
3. Integral:
Solution:
4. Integral: dx
Solution:
What are the formulas for ?
Solve these integrals
1.
2.
3.
4.
Additional Problems
1.
Hint: What is u? What is du? How can I use substitution to rewrite this integral so that the
power rule for integration can be used?
2.
Hint: Is there an identity I can use to rewrite this problem so a basic trig integration formula
can be used?
3.
Hint: What is u? What is du? How can I use substitution to rewrite this integral so that the
power rule for integration can be used?
4.
Hint: How can we break this integral apart into two less complicated problems?
