1) Integrate (antidifferentiate, ie the opposite process of finding a derivative) the given function for dy/dx:
y = 1/2x^2 +sinx + C. (C is a #, called the constant of integration, and is included in any indefinite integral.)
(Notice you can verify by taking y's derivative that it gives the correct dy/dx.)
Finally, solve for the constant, C, using the other given, y(3pi/4) = 4:
y(3pi/4) = 1/2 (3pi/4)^2 + sin(3pi/4) + C = 4. So C = 4 - sqrt2/2 - 9pi^2/32 (this is a really ugly number)
2) Multiply out the function being integrated: x^3 +3x^2 - 10x
Integrate using reverse power rule: 1/4x^4 + x^3 - 5x^2 ] 2 0
Evaluate the integral function at 2 and at 0. Subtract the second answer from the first:
-8 - 0 = - 8
3) This is called a Riemann sum, and it is just adding up the areas of rectangles. Left endpoints means you will use the height (y-value) of the function at the left of each rectangle. n = 3 means 3 rectangles (each of width 2)
So the heights you will use are at x = 0, x = 2, and x = 4. Those heights are 0, 4, and 16. The combined areas are 0*2 + 4*2 + 16*2 = 40. This will significantly UNDERestimate the actual area (because the curve is concave down).
The actual area is given by the definite integral of x^2 on the closed interval [0,6]:
1/3x^3 ] 6 0 = 216/3 - 0 = 72 The riemann sum estimate was off by 12 sq. units or 16.67%.
