Exam 5: Integration

Use left endpoints and 10 rectangles to find the approximation of the area of the region between the graph of the function and the x-axis over the interval [2, 13]. Round your answer to the nearest integer. ​

Find the sum given below. ​ ​

Find the indefinite integral . ​

Use sigma notation to write the sum . ​

Find if . ​

The height above the ground of an object thrown upward from a point feet above the ground with an initial velocity of feet per second is given by the function . A balloon, rising vertically with a velocity of 8 feet per second, releases a sandbag at the instant it is 44 feet above the ground. At what velocity will it hit the ground? Round your answer to three decimal places. ​

The force F (in newtons) of a hydraulic cylinder in a press is proportional to the square of where x is the distance (in meters) that the cylinder is extended in its cycle. The domain of F is and . Find F as a function of x. ​

The volume V in liters of air in the lungs during a five-second respiratory cycle is approximated by the model where t is the time in seconds. Approximate the average volume of air in the lungs during one cycle. Round your answer to four decimal places.

Find given . ​

Find the indefinite integral. ​ ​

Evaluate . Round your answer to three decimal places. ​

Find . ​

Sketch the region whose area is given by the definite integral and then use a geometric formula to evaluate the integral. ​ ​

Write the limit , as a definite integral on the interval [4, 6], where is any point in the subinterval. ​

Find the integral . ​

Evaluate the definite integral of the algebraic function. ​ ​ Use a graphing utility to verify your results. ​

Find the area of the shaded region for the function . ​ ​

Evaluate the definite integral . Use a graphing utility to verify your results. ​

Find the average value of on the interval [1, 3]. ​

Find the area of the region bounded by the graphs of the equations . Round your answer to the nearest whole number. ​