site.iugaza.edu.pssite.iugaza.edu.ps/selshamy/files/2010/02/MathA_Ch53.pdfTHEOREM I—lnlegrability of Continuous Functions In this chapter we develop a method calculate the areas

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Example 4. Find 𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑

𝑖𝑖𝑖𝑖 𝑑𝑑 = ∫ 𝑠𝑠𝑠𝑠𝑠𝑠2𝑡𝑡𝑡𝑡𝑡𝑡𝑑𝑑𝑑𝑑 𝑡𝑡 𝑑𝑑𝑡𝑡.

Example 5.

𝐼𝐼𝑖𝑖 𝑖𝑖(𝑑𝑑) = ∫ √1 + 𝑡𝑡2 𝑡𝑡𝑡𝑡𝑡𝑡𝑑𝑑1 𝑑𝑑𝑡𝑡 𝐹𝐹𝑖𝑖𝑡𝑡𝑑𝑑 𝑖𝑖 �𝜋𝜋

4� 𝑡𝑡𝑡𝑡𝑑𝑑 𝑖𝑖′(𝜋𝜋

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Example 6. Find f(x) and a if

2∫ 𝑖𝑖(𝑡𝑡)𝑑𝑑𝑡𝑡 = 2𝑠𝑠𝑖𝑖𝑡𝑡𝑑𝑑 − 1, 0 ≤ 𝑡𝑡 ≤ 𝜋𝜋2

𝑑𝑑𝑡𝑡 .

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Remark:

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Examples:

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Examples:

1) Find the area of the region in the first quadrant bounded above by y = 1 below by y = x , y = x2/4 .

3) Find the area enclosed by x = y2 , x = y +2 .