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Example Questions
Example Question #1 :How To Find Area Of A Region
What is the average value of the function f(x) = 12x3+ 15x + 5 on the interval [3, 6]?
1542
1350.2
1302.5
771
895.67
1302.5
To find the average value, we must take the integral of f(x) between 3 and 6 and then multiply it by 1/(6 – 3) = 1/3.
The indefinite form of the integral is: 3x4+ 7.5x2+ 5x
The integral from 3 to 6 is therefore: (3(6)4+ 7.5(6)2+ 5(6)) - (3(3)4+ 7.5(3)2+ 5(3)) = (3888 + 270 + 30) – (243 + 22.5 + 15) = 3907.5
The average value is 3907.5/3 = 1302.5
Example Question #1 :How To Find Area Of A Region
Find the dot product ofa= <2,2,-1> andb= <5,-3,2>.
To find the dot product, we multiply the individual corresponding components and add.
Here, the dot product is found by:
2 * 5 + 2 * (-3) + (-1) * 2 = 2.
Example Question #3 :How To Find Area Of A Region
Find the area of the region enclosed by the parabolaand the line.
The limits of the integration are found by solvingandfor:
The region runs fromto. The limits of the integration are,.
The area between the curves is:
Example Question #4 :How To Find Area Of A Region
Find
Example Question #1 :How To Find Area Of A Region
Find
Example Question #6 :How To Find Area Of A Region
What is the area of the space belowand above
is only aboveover the interval. Areas are given by the definite integral of each functionand
The area between the curves is found by subtracting the area between each curve and the-axis from each other. Forthis area isand forthe area isgiving an area between curves of
Example Question #7 :How To Find Area Of A Region
What is the area belowand above the-axis?
To find the area below a curve, you must find the definite integral of the function. In this case the limits of integration are where the original function intercepts the-axis atand. So you must findwhich isevaluated fromto. This gives an answer of
Example Question #8 :How To Find Area Of A Region
Find the area between the curvesand.
To solve this problem, we first need to find the point where the two equations are equal. Doing this we find that
.
From this, we see that the two graphs are equal atand. We also know that for,is greater than.
So to find the area between these curves we need to evaluate the integral.
The solution to the integral is
.
Evaluating this atandwe get
Example Question #9 :How To Find Area Of A Region
Find the value of
To solve this problem, we will need to do a-substitution. Letting
.
Substituting ourfunction back into the integral, we get
Evaluating this atandwe get
Example Question #10 :How To Find Area Of A Region
Find the average value ofon the interval
The average is given by integration as:
This means that:
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