Presentation on theme: "Review Problem – Riemann Sums Use a right Riemann Sum with 3 subintervals to approximate the definite integral:"— Presentation transcript:
Review Problem – Riemann Sums Use a right Riemann Sum with 3 subintervals to approximate the definite integral:
Applications of the Definite Integral Mr. Reed AP Calculus AB
Finding Areas Bounded by Curves To get the physical area bounded by 2 curves: 1.Graph curves & find intersection points – limits of integration 2.Identify “top” curve & “bottom” curve OR “right-most” curve & “left-most” curve 3.Draw a representative rectangle 4.Set up integrand: Top – Bottom Right – Left
Finding Intersection Points Set equations equal to each other and solve algebraically Graph both equations and numerically find intersection points
Example #1 Find the area of the region between y = sec 2 x and y = sinx from x = 0 to x = pi/4.
Example #2 Find the area that is bounded between the horizontal line y = 1 and the curve y = cos 2 x between x = 0 and x = pi.
Example #3 From Text – p #16
Example #4 Find the area of the region R in the first quadrant that is bounded above by y = sqrt(x) and below by the x-axis and the line y = x – 2.
Summarize the process
AP MC Area Problem #12 from College Board Course Description
Homework P : Q1-Q10, 13-25(odd)
Authentic Applications for the Definite Integral Example #2 – p.237
Definite Integral Applied to Volume 2 general types of problems: 1.Volume by revolution 2.Volumes by base
Volume by Revolution – Disk Method The region under the graph of y = sqrt(x) from x = 0 to x = 2 is rotated about the x-axis to form a solid. Find its volume.
Volume by Revolution – Disk Method
Homework #1 – Disk Method about x and y axis P : Q1-Q10,1,3,5
Volume by Revolution – About another axis The region bounded by y = 2 – x^2 and y = 1 is rotated about the line y = 1. Find the volume of the resulting solid.
Volume by Revolution – Washer Method Find the volume of the solid formed by revolving the region bounded by the graphs of f(x) = sqrt(x) and g(x) = 0.5x about the x-axis.