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Applications of the Integral

This document presents multiple examples demonstrating the application of integrals in finding the areas of regions bounded by various curves. Each example guides you through the process of determining the intersections of curves and setting up the appropriate integrals. For instance, we examine areas defined by curves such as (y = x^2 + 1) and (y = x), as well as the intersection points of trigonometric functions. Additional examples cover graphical analysis in the first quadrant with specific curves providing a comprehensive understanding of integral applications in geometry.

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Applications of the Integral

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  1. Applications of the Integral

  2. Areas

  3. Example (1)Find the area A of the region bounded by:y=x2 +1, y=x , x=0 & x=1.

  4. Example (2)Find the area A of the region bounded by:y=x2, y=2x-x2

  5. Intersections of the Two Curves

  6. Example (3)Find the area A of the region bounded by:y=cosx, y=sinx, x=0 and x=π/2

  7. Intersection of the Two Graphs

  8. Example (4)Find the area A of the region in the first quadranbounded by:y=x2+ 1 and y-10=0 We second curve’s equation can be rewritten as: y = 10 The curves intersects at the points x=3 and x=-3. We arrive at that by letting x2+ 1=10, which leads to x2=9 The following figures shows the mentioned region. Find the area using two methods!

  9. Example (5)Set up the integral/integrals to find the area A of the region bounded by:y=9-x2, y=x2 + 1 from x=-4 to x=3

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