Question 1

The manager of TeleStar Cable Service estimates that the total number of subscribers to the service in a certain city t years from now will be
​  ​
Find the average number of cable television subscribers over the next 8 years if this prediction holds true. Please round your answer to the nearest whole number.

Accepted Answer

The answer of The manager of TeleStar Cable Service estimates...

Question 2

Evaluate the given definite integral.
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Accepted Answer

The answer of Evaluate the given definite integral.
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Question 3

Evaluate the given definite integral.
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Accepted Answer

The answer of Evaluate the given definite integral.
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Question 4

Evaluate the given definite integral.
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Accepted Answer

The answer of Evaluate the given definite integral.
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Question 5

Evaluate the given definite integral.
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Accepted Answer

The answer of Evaluate the given definite integral.
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Question 6

The price of a certain commodity in dollars/unit at time t (measured in weeks) is given by
​  ​
What is the average price of the commodity over the 6-week period from t = 0 to t = 6? Please round the answer to the nearest cent.
​
$__________

Accepted Answer

The answer of The price of a certain commodity in...

Question 7

Evaluate the given definite integral.
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Accepted Answer

The answer of Evaluate the given definite integral.
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Question 8

Evaluate the definite integral.
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Accepted Answer

The answer of Evaluate the definite integral.
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Question 9

Evaluate the definite integral.
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Accepted Answer

The answer of Evaluate the definite integral.
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Question 10

Evaluate the given definite integral.
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Accepted Answer

The answer of Evaluate the given definite integral.
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Question 11

Evaluate the given definite integral.
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Accepted Answer

The answer of Evaluate the given definite integral.
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Question 12

A bottle of white wine at room temperature (68°F) is placed in a refrigerator at 5 P.M. Its temperature after t hours is changing at the rate of
​  ​
°F/hr. By how many degrees will the temperature of the wine have dropped by 11 P.M.? Please round your answer to one decimal place.
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__________ °F
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What will the temperature of the wine be at 11 P.M.? Please round your answer to one decimal place.
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__________ °F

Accepted Answer

The answer of A bottle of white wine at room...

Question 13

Find the average value of the function
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over the indicated interval  .

Accepted Answer

The answer of Find the average value of the function
​...

Question 14

Find the area of the region under the graph of the function ​  ​
On the interval  .
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Accepted Answer

The answer of Find the area of the region under...

Question 15

Evaluate the definite integral.
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Accepted Answer

The answer of Evaluate the definite integral.
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Question 16

Evaluate the given definite integral.
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Accepted Answer

The answer of Evaluate the given definite integral.
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Question 17

Find the average value of the function
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over the interval  .

Accepted Answer

The answer of Find the average value of the function
​...

Question 18

Find the area of the region under the graph of the function ​  ​
On the interval  , using the fundamental theorem of calculus. Then verify your result using geometry.
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Accepted Answer

Using the fundamental theorem of calculus, we can find the area under the graph of the function by taking the antiderivative of the function and evaluating it at the endpoints of the interval:

$$\int_{-2}^{1} (2x+2) dx = \left[x^2 + 2x\right]_{-2}^{1} = (1^2+2\cdot1)-((-2)^2+2\cdot(-2)) = 3+8 = 11$$

So the area under the graph is 11 sq units.

To verify this result using geometry, we can graph the function and the interval and find the area of the corresponding trapezoid:

[asy]
unitsize(0.45 cm);

pair[] A,B;
real f(real x)
{
  return((x^2+2*x-7)/2);
}

A[0] = (-2,0);
B[0] = (1,0);
A[1] = A[0];
B[1] = (A[1].x,f(A[1].x));
A[2] = B[0];
B[2] = (B[0].x,f(B[0].x));
A[3] = B[1];
B[3] = B[2];

fill(B[1]--B[2]--(B[2].x,0)--(B[1].x,0)--cycle,gray(0.7));
draw(A[0]--B[0],dashed);
draw(B[0]--B[3]);
draw(A[1]--B[1]);
draw(A[2]--B[2]);
draw((-2.5,0)--(2.5,0));
draw((0,-2)--(0,16));
label("$y = \frac{1}{2}(x^2+2x-7)$", (2,7));
label("$(-2,0)$", A[0], S);
label("$(1,0)$", B[0], S);
[/asy]

The height of the trapezoid is the maximum value of the function on the interval, which occurs at $x = -1$, where $y = \frac{1}{2}((-1)^2+2(-1)-7) = -\frac{5}{2}$. The lengths of the top and bottom sides are $f(-2) = \frac{1}{2}((-2)^2+2(-2)-7) = -\frac{9}{2}$ and $f(1) = \frac{1}{2}(1^2+2(1)-7) = -\frac{3}{2}$, respectively. Therefore, the area of the trapezoid is:

$$\frac{1}{2} \cdot (-\frac{9}{2} - \frac{3}{2}) \cdot (-\frac{5}{2}) = \frac{12}{4} \cdot \frac{5}{2} = 15$$

This is not equal to the value obtained using the fundamental theorem of calculus, so we made an error in our geometry calculation. To find the correct area using geometry, we can divide the trapezoid into a rectangle and a right triangle:

[asy]
unitsize(0.45 cm);

draw((-2.5,0)--(2.5,0));
draw((0,-2)--(0,16));
label("$y = \frac{1}{2}(x^2+2x-7)$", (2,7));

fill((-2,0)--(-2,-7)--(1,-2)--(1,0)--cycle,gray(0.7));
draw((-2,-7)--(-2,0)--(1,0)--(1,-2));
draw((-2,-7)--(1,-7),dashed);
draw((1,-2)--(2,-2),dashed);
draw((1,-2)--(1,0),dashed);
draw(rightanglemark((1,-2),(1,0),(2,0),10));

label("$(-2,0)$", (-2,0), S);
label("$(1,0)$", (1,0), S);
label("$(1,-2)$", (1,-2), E);
label("$(-2,-7)$", (-2,-7), W);
label("$(2,-2)$", (2,-2), E);
[/asy]

The area of the rectangle is $(1-(-2))\cdot(-2) = 6$, and the area of the right triangle is $\frac{1}{2} \cdot (1-(-2)) \cdot (-7-(-2)) = \frac{15}{2}$. Therefore, the total area under the graph is:

$$6 + \frac{15}{2} = \frac{12+15}{2} = \frac{27}{2} = 13.5 \approx 14$$

This is closest to option C, 12 sq units. However, none of the answer choices match the exact value of either the calculus or geometry calculation.

Question 19

Evaluate the given definite integral.
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Accepted Answer

The answer of Evaluate the given definite integral.
​ ...

Question 20

Find the area of the region under the graph of the function ​  ​
On the interval  , using the fundamental theorem of calculus. Then verify your result using geometry.
​

Accepted Answer

The answer of Find the area of the region under...

Quiz 6: Integration

Evaluate the given definite integral.

Evaluate the given definite integral.

Evaluate the given definite integral.

Evaluate the given definite integral.

The price of a certain commodity in dollars/unit at time t (measured in weeks) is given by What is the average price of the commodity over the 6-week period from t = 0 to t = 6? Please round the answer to the nearest cent. $__________

Evaluate the given definite integral.

Evaluate the definite integral.

Evaluate the definite integral.

Evaluate the given definite integral.

Evaluate the given definite integral.

Find the average value of the function over the indicated interval .

Find the area of the region under the graph of the function On the interval .

Evaluate the definite integral.

Evaluate the given definite integral.

Find the average value of the function over the interval .

Find the area of the region under the graph of the function On the interval , using the fundamental theorem of calculus. Then verify your result using geometry.

Evaluate the given definite integral.

Find the area of the region under the graph of the function On the interval , using the fundamental theorem of calculus. Then verify your result using geometry.

Quiz 6: Integration

Evaluate the given definite integral. ​

Evaluate the given definite integral. ​

Evaluate the given definite integral. ​

Evaluate the given definite integral. ​

The price of a certain commodity in dollars/unit at time t (measured in weeks) is given by ​ ​ What is the average price of the commodity over the 6-week period from t = 0 to t = 6? Please round the answer to the nearest cent. ​ $__________

Evaluate the given definite integral. ​

Evaluate the definite integral. ​

Evaluate the definite integral. ​

Evaluate the given definite integral. ​

Evaluate the given definite integral. ​

Find the average value of the function ​ ​ over the indicated interval .

Find the area of the region under the graph of the function ​ ​ On the interval . ​

Evaluate the definite integral. ​

Evaluate the given definite integral. ​

Find the average value of the function ​ ​ over the interval .

Find the area of the region under the graph of the function ​ ​ On the interval , using the fundamental theorem of calculus. Then verify your result using geometry. ​

Evaluate the given definite integral. ​

Find the area of the region under the graph of the function ​ ​ On the interval , using the fundamental theorem of calculus. Then verify your result using geometry. ​

Evaluate the given definite integral.

Evaluate the given definite integral.

Evaluate the given definite integral.

Evaluate the given definite integral.

The price of a certain commodity in dollars/unit at time t (measured in weeks) is given by What is the average price of the commodity over the 6-week period from t = 0 to t = 6? Please round the answer to the nearest cent. $__________

Evaluate the given definite integral.

Evaluate the definite integral.

Evaluate the definite integral.

Evaluate the given definite integral.

Evaluate the given definite integral.

Find the average value of the function over the indicated interval .

Find the area of the region under the graph of the function On the interval .

Evaluate the definite integral.

Evaluate the given definite integral.

Find the average value of the function over the interval .

Find the area of the region under the graph of the function On the interval , using the fundamental theorem of calculus. Then verify your result using geometry.

Evaluate the given definite integral.

Find the area of the region under the graph of the function On the interval , using the fundamental theorem of calculus. Then verify your result using geometry.