# 1 4 using four rectangles of width 1 which touch the

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4. Using four rectangles, of width 1, which touch the curve on the left, estimate the area under the graph of y = x 3 + 1 from x = 0 to x = 4. 5. Redo problem 4 but now the rectangles touch the curve on the right. 6. Reredo problem 4 but now the rectangles touch the curve at the midpoint of the top side. 7. Rereredo problem 4 but with eight rectangles (of width 1 / 2) 8. Rerereredo problem 4 but with 12 rectangles (this may take a while). 9. Find the exact area beneath the curve (in the same interval). Does having more rectangles lead to a better estimate? 2
10. Find the general antiderivative of f ( x ) = 0. 11. Find an antiderivative of f ( x ) = 32 x - 76. 12. Find an antiderivative of f ( x ) = 8 x 3 + x . 13. Find the general antiderivative of f ( x ) = 2 x . 14. Find an antiderivative of x . 15. Consider the following graph (the blue line): - 4 - 3 - 2 - 1 1 2 3 4 - 4 - 3 - 2 - 1 1 2 3 4 x y Find the integral of it from - 4 to 2 (i.e. if f ( x ) is the blue graph, evaluate Z 2 - 4 f ( x ) dx ). 3
16. Now find the integral of the above graph from - 4 to 4. 17. State the power rule for antiderivatives (i.e. what is the antiderivative of x n ?) 18. What is the area under the curve of the graph y = 9 - x 2 and above the x-axis? 19. What is the area above the curve y = | x 3 | - 8 and below the x-axis? 20. Evaluate Z 3 » 4 - ( x - 1) 2 dx (hint: draw the graph first) 1 21. What is the area above the curve y = x 2 + 4 x + 5 and below the x-axis? 22. Who invented the integral symbol? 23. (Bonus Problem!) Evaluate this integral: Z 1001 1 x 2 ( x 3 - 1) - 2 / 3 . 4