Question 1

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...

Question 2

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

Accepted Answer

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

Question 3

Find the average value of the function  over the interval  .

Accepted Answer

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

Question 4

A bottle of white wine at room temperature (  ) is placed in a refrigerator at 4 P.M.Its temperature after t hours is changing at the rate of   .By how many degrees will the temperature of the wine have dropped by 8 P.M.? Please round your answer to one decimal place.
__________  What will the temperature of the wine be at 8 P.M.? Please round your answer to one decimal place.
__________

Accepted Answer

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

Question 5

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

Accepted Answer

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

Question 6

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

Accepted Answer

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

Question 7

Evaluate the definite integral.

Accepted Answer

The answer of Evaluate the definite integral. 
...

Question 8

Evaluate the definite integral.

Accepted Answer

The answer of Evaluate the definite integral. 
...

Question 9

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...

Question 10

Evaluate the definite integral.

Accepted Answer

The answer of Evaluate the definite integral. 
...

Question 11

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

Accepted Answer

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

Question 12

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 fundamental theorem of calculus states that if $f(x)$ is a continuous function on $[a,b]$, then the area of the region under the graph of $f(x)$ on the interval $[a,b]$ is given by $$\int_a^b f(x) dx.$$ Applying this theorem to the given function, we have $$\int_{-6}^{4} 3x^2-2x+1 \ dx = \left[x^3-x^2+x\right]_{-6}^{4} = (64-16+4)-(216-36-6) = \boxed{196}.$$ To verify this using geometry, we can graph the function and compute the area directly as the area of a trapezoid with base lengths 10 and 2 and height 16. This gives us an area of $\frac{(10+2)16}{2} = \boxed{96}$, which is not equal to the result we obtained using the fundamental theorem of calculus. Therefore, option B is the only correct choice.

Question 13

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

Accepted Answer

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

Question 14

Evaluate the definite integral.

Accepted Answer

The answer of Evaluate the definite integral. 
...

Question 15

Find the area of the region under the graph of the function f on the interval [a, b].

Accepted Answer

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

Question 16

Evaluate the definite integral.

Accepted Answer

The answer of Evaluate the definite integral. 
...

Question 17

Evaluate the given definite integral.

Accepted Answer

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

Question 18

Evaluate the definite integral.

Accepted Answer

The answer of Evaluate the definite integral. 
...

Question 19

Evaluate the definite integral.

Accepted Answer

The answer of Evaluate the definite integral. 
...

Question 20

Evaluate the definite integral.

Accepted Answer

The answer of Evaluate the definite integral. 
...

Quiz 6: Integration

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.

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

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 .

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

Evaluate the definite integral.

Evaluate the 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 definite integral.

Find the area of the region under the graph of the function on 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.

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

Evaluate the definite integral.

Find the area of the region under the graph of the function f on the interval [a, b].

Evaluate the definite integral.

Evaluate the given definite integral.

Evaluate the definite integral.

Evaluate the definite integral.

Evaluate the definite integral.