(a) integrate to find F as a function of x, and (b) demonstrate the Second Fundamental Theorem of Calculus by differentiating the result in part (a). $F(x)=\int_{4}^{x} t^{3 / 2} d t$

Evaluate the definite integral. 2π∫_0^1 (y+1)√(1-y) dy

At the instant the traffic light turns green, a car that has been waiting at an intersection starts with a constant acceleration of 6 feet per second per second. At the same instant, a truck traveling with a constant velocity of 30 feet per second passes the car. (a) How far beyond its starting point will the car pass the truck? (b) How fast will the car be traveling when it passes the truck?

Find the area of the region. ∫ _π/2^2π/3 [sec²(x/2) dx]

Prove that

$$∫_a^b x dx = b² - a² / 2.$$

In Exercise given below, demonstrate the Second Fundamental Theorem of Calculus by differentiating the result in part (a).

$$F(x)=\int_0^x t\left(t^2+1\right) d t$$

(a) integrate to find as a function of , and (b) demonstrate the Second Fundamental Theorem of Calculus by differentiating the result in part (a). F(x) = ∫_0^x t(t² + 1) dt

Evaluate the definite integral. ∫_3^6 x/3√(x²-8) dx

Find the area of the region. ∫_(-2)^6 [x²∛(x+2) dx]

Does the Midpoint Rule ever give the exact area between a function and the x-axis? Explain.

Find the Riemann sum for f(x) = sin x over the interval [0, 2π], where x_0 = 0, x_1 = π/4, x_2 = π/3, x_3 = π, and x_4 =2π, and where c_1 = π/6, c_2 = π/3, c_3 = 2π / 3, and c_4 = 3π/2.

A car traveling at 45 miles per hour is brought to a stop, at constant deceleration, 132 feet from where the brakes are applied. (a) How far has the car moved when its speed has been reduced to 30 miles per hour? (b) How far has the car moved when its speed has been reduced to 15 miles per hour? (c) Draw the real number line from 0 to 132. Plot the points found in parts (a) and (b). What can you conclude?

Find the indefinite integral. ∫ (2x-9 sin x) dx