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c Math 152, Benjamin Aurispa 7.1 Area Between Curves If y = f (x) and y = g(x) are continuous functions and if f (x) ≥ g(x) on an interval a ≤ x ≤ b, then the area of the region bounded by the curves on that interval is Z b [f (x) − g(x)] dx a If x = f (y) and x = g(y) are continuous functions and if f (y) ≥ g(y) on an interval a ≤ y ≤ b, then the area of the region bounded by the curves on that interval is Z b a [f (y) − g(y)] dy Examples: Find the area of the region(s) bounded by the curves. • y = 5x − x2 , y = 2x − 4 1 c Math 152, Benjamin Aurispa • x = y 2 , x − 2y = 3 • y = |x| − 1, y = 5 − x2 • x = 10 − y 2 , x = 2y 2 − 5 2 c Math 152, Benjamin Aurispa • y = x2 + 4, y = 2x2 , from x = −2 to x = 4. • y = 1 − sin x, y = sin x from x = 0 to x = π 2 3 c Math 152, Benjamin Aurispa Set up the integral(s) to find the area bounded by the curves y = 1/x2 , y = x, and y = 18 x. Find the area of the region bounded by the curve y = 2x2 , the tangent line to this curve at the point (3, 18), and the x-axis. 4 c Math 152, Benjamin Aurispa 7.2 Volumes General Principle: Disks About x-axis: About y-axis: 5 c Math 152, Benjamin Aurispa Examples: Find the volumes of the solids found by rotating the regions bounded by the given curves about the specified axis/line. √ • y = 2 x, y = 0, x = 9 about the x-axis • y = x3 , y = 8, x = 0 about the y-axis • y = (x − 2)2 , y = 9 about the line y = 9 6 c Math 152, Benjamin Aurispa • x = (y − 1)2 + 2, x = 6 about the line x = 6 Washers Examples: Find the volumes of the solids found by rotating the regions bounded the given curves about the specified axis/line. • y = x2 + 2, y = 3x about the x-axis • Same region about the y-axis 7 c Math 152, Benjamin Aurispa • x = y 2 , y = x3 about the line y = 3 • Same region about the line x = −2 •y= √ x + 1, y = 1, x = 4 about the line x = 8 8 c Math 152, Benjamin Aurispa Same region about the line y = −5: Volumes by Cross-Sections Example where base is a circle and cross-sections perpendicular to the x-axis are equilateral triangles. We’ll do a problem like this in the third example below. First, we’ll do a few ”easier” ones. Example 1: Find the volume of the solid S whose base is the region bounded by the curves y = x2 and y = 4. Cross sections perpendicular to the y-axis are squares. 9 c Math 152, Benjamin Aurispa Example 2: Find the volume of the solid S whose base is the triangular region with endpoints (0, 0), (6, 0), and (0, 2). Cross-sections perpendicular to the x-axis are semicircles. Example 3: Find the volume of the solid S whose base is the circle x2 + y 2 = 9. Cross sections perpendicular to the x-axis are equilateral triangles. 10