Topics 6.1-6.6 Review

A child is inflating a balloon by blowing air in to it. Each time he takes a breath, a small amount of air leaks out of the balloon. The table gives various values for time t (in seconds) and r'(t), the rate of change of the radius of the balloon in mm/sec. Use a Left Riemann Sum with three subintervals to approximate the total change in radius of the balloon on the interval [1,11].

• 14

• 16

• 27

• 38

A function g(x) has g'(x)>0 and g"(x)<0 on [1,5]. If a Right Riemann Sum is used to approximate the area between g(x) and the x-axis on the interval [1, 5], which of the following is correct?

• underapproximate because g'(x)>0

• overapproximate because g'(x)>0

• underapproximate because g"<0

• overapproximate because g"<0

Samantha is heating up some soup on the stove and various measurements of temperature, in degrees Celsius, are taken and given in the table above. Using a midpoint Riemann sum with three subintervals, which of the following expressions could be used to find the total degrees the soup accumulated over the interval [0,8]?

•  $\left[2\left(22\right)+2\left(35\right)+4\left(51\right)\right]$  

•  $\left[1\left(22\right)+3\left(35\right)+6\left(51\right)\right]$  

•  $\left[2\left(26\right)+2\left(48\right)+4\left(62\right)\right]$  

•  $\left[2\left(23\right)+2\left(37\right)+4\left(55\right)\right]$  

Which of the following limits is equal to

 $\int_1^3x^3dx$  

•  $\lim_{n\rightarrow\infty}\sum_{k=1}^n\left(1+\frac{k}{n}\right)^3\frac{1}{n}$  

•  $\lim_{n\rightarrow\infty}\sum_{k=1}^n\left(1+\frac{2k}{n}\right)^3\frac{2}{n}$  

•  $\lim_{n\rightarrow\infty}\sum_{k=1}^n\left(1+\frac{k}{n}\right)^3\frac{2}{n}$  

•  $\lim_{n\rightarrow\infty}\sum_{k=1}^n\left(1+\frac{2k}{n}\right)^3\frac{1}{n}$  

Gasoline is being pumped into a car. The rate that the gas is being pumped is given in the table at selected times. Use a right Riemann Sum with 3 equal subintervals to approximate the total gallons of gasoline pumped in the car over the 24 seconds.

• 8.72 gallons

• 9.92 gallons

• 6.96 gallons

• 7.51 gallons

Gasoline is being pumped into a car. The rate that the gas is being pumped is given in the table at selected times. Use a midpoint Riemann Sum with 3 equal subintervals to approximate the total gallons of gasoline pumped in the car over the 24 seconds.

• 8.72 gallons

• 9.92 gallons

• 6.96 gallons

• 7.51 gallons

Gasoline is being pumped into a car. The rate that the gas is being pumped is given in the table at selected times. Use a left Riemann Sum with 3 equal subintervals to approximate the total gallons of gasoline pumped in the car over the 24 seconds.

• 8.72 gallons

• 9.92 gallons

• 6.96 gallons

• 7.51 gallons

The rate that people are entering a local office is given in the table. Use a left Riemann sum with 4 subintervals to approximate the total number of people entering the office over the interval [0,7].

• 44 people

• 42 people

• 43 people

• 41 people

The rate that people are entering a local office is given in the table. Use a trapezoidal sum with 4 subintervals to approximate the total number of people entering the office over the interval [0,7].

• 44 people

• 42 people

• 43 people

• 41 people

Translate the definite integral into a Riemann sum.

 $\int_1^3\left(x+1\right)dx$  

•  $\lim_{n\rightarrow\infty}\sum_{k=1}^n\left[\left(1+\frac{2k}{n}\right)+1\right]\left(\frac{2}{n}\right)$  

•  $\lim_{n\rightarrow\infty}\sum_{k=1}^n\left[\left(2+\frac{k}{n}\right)+1\right]\left(\frac{1}{n}\right)$  

•  $\lim_{n\rightarrow\infty}\sum_{k=1}^n\left[\left(1+\frac{3k}{n}\right)+1\right]\left(\frac{3}{n}\right)$  

•  $\lim_{n\rightarrow\infty}\sum_{k=1}^n\left[\left(1+\frac{2k}{n}\right)+3\right]\left(\frac{2}{n}\right)$