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
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