Exam 5: Integration

Answer true or false: The average value of f(x) = x⁷ on the interval [-2, 2] is 12.

Find F'(x) if .

Find the distance an object travels if its velocity is v(t) = t - 6t² in m/s, where 0 $\le$ t $\le$ 1.

Find the area under the curve y = x² + 8 for 2 $\le$ x $\le$ 7.

Given f(x) = 8 + x . Use the rectangle method to approximate the area on the interval [0, 6] using 4 rectangles. (Assume the graph goes through the midpoint of each rectangle.)

Evaluate .

Find the area under the curve over the interval [0, 5].

Find the distance an object travels if its velocity is v(t) = t - 9t² in m/s, where 0 $\le$ t $\le$ 3.

Answer true or false. $\int$ x²(x³ + 1)^2/3 dx can be easily solved by letting u = x³ + 1.

Find the derivative. .

A stone is thrown downward from the top of a 1008 foot high cliff with an initial velocity of 32 feet per second. What is the speed of the stone upon impact with the ground?

Evaluate .

Find the average value of f(x) = e ^x - 1 over the interval [-1, 1]. Approximate value to three decimal places.

Evaluate .

Evaluate by first changing f(k) = (k + 4)³ to f(i) = i³ and then by making an appropriate change in the limits of summation.

Write the expression in sigma notation. x⁷ + x⁸ + x⁹ + x¹⁰ + x¹¹.

A particle moves with a velocity of v(t) meters per second along an s-axis. Find the displacement that the particle travels during the period 0 $\le$ t $\le$ 2. v(t) = e^2t .

A projectile is launched vertically upward from the ground with an initial velocity of 144 feet per second. How long does it take the projectile to reach the ground?

Evaluate by first changing f(k) = (k + 3)² to f(i) = i² and then, an appropriate change in the limits of summation.

Answer true or false. For a good choice for u is 13- x².