Let f(x) = 3e−x. Let R be the region in the plane bounded by the curves y = f(x), y = 4e−3, and x = 1. For all of the integrals below, do not evaluate the definite integrals until asked to do so numerically using Desmos. i) Write down a definite integral that gives the area of R. ii) Write down a definite integral that gives the volume of the region, when R is rotated around the x-axis. iii) Write down a definite integral that gives the volume of the region, when R is rotated around the line y=-2.
Let f(x) = 3e−x. Let R be the region in the plane bounded by the curves y = f(x), y = 4e−3, and x = 1. For all of the integrals below, do not evaluate the definite integrals until asked to do so numerically using Desmos. i) Write down a definite integral that gives the area of R. ii) Write down a definite integral that gives the volume of the region, when R is rotated around the x-axis. iii) Write down a definite integral that gives the volume of the region, when R is rotated around the line y=-2.
Let f(x) = 3e−x. Let R be the region in the plane bounded by the curves y = f(x), y = 4e−3, and x = 1. For all of the integrals below, do not evaluate the definite integrals until asked to do so numerically using Desmos. i) Write down a definite integral that gives the area of R. ii) Write down a definite integral that gives the volume of the region, when R is rotated around the x-axis. iii) Write down a definite integral that gives the volume of the region, when R is rotated around the line y=-2.
Let f(x) = 3e−x. Let R be the region in the plane bounded by the curves y = f(x), y = 4e−3, and x = 1. For all of the integrals below, do not evaluate the definite integrals until asked to do so numerically using Desmos. i) Write down a definite integral that gives the area of R. ii) Write down a definite integral that gives the volume of the region, when R is rotated around the x-axis. iii) Write down a definite integral that gives the volume of the region, when R is rotated around the line y=-2. iv) Write down a definite integral that gives the volume of the region, when R is rotated around the y-axis. v) Write down a definite integral that gives the volume of the region, when R is rotated around the line x = −2. vi) Write down a definite integral that gives the arc length of the curve y = f(x) over the region that bounds the top of R. vii) Write down a definite integral that gives the surface area of the outer curved part of the surface, when R is rotated around the x-axis. viii) Now, using the built-in numerical integration feature of Desmos, evaluate all of these integrals numerically.
With differentiation, one of the major concepts of calculus. Integration involves the calculation of an integral, which is useful to find many quantities such as areas, volumes, and displacement.
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