base of a solid in the xy-plane is the 1st quadrant region bounded y=x&y=x^2cross section of solid perpendicular to x-axis are equilateral triangles. whats the volume,in cubic units of solid? -Determine if Mean Value Theorem for integrals applies to the function f(x)=x^3-9x on the interval [-1,1]if so find x-coordinates of points guaranteed to exist by theorem.-object has a constant acceleration of 40 ft/sec^2,an inital velocity of -20ft/sec,&inital posit of10ft find function s(t)descrbe motion.
graphs y = x and y = x2
intersect at the points (0, 0) and (1, 1). So, V = ∫01
A(x)dx, where A(x) is the area of the cross-section of the solid by a
plane perpendicular to the x-axis. Since the area of an equilateral
triangle with the side a equals A = (√(3)/4)a2,
the function A(x) = (√(3)/4)(x
and the volume V = ∫01
– 2x3 +
– 1/2 + 1/5) = √(3)/120.
to the Mean Value Theorem ∫–11(x3
– 9x) dx = 2f(x*), where x* lies between –1 and 1. However, the
integral equals 0 (the integrand is an odd function and the interval
of integration is symmetric with respect to 0), thus, f(x*) = 0. The
only possible value of x* is 0 (x3
– 9x = 0 has also solutions ±3 but they do not lie between –1
law of motion can be described as s(t) = s0
= 10 – 20t + 20t2,
where time t is expressed in seconds and the position s in feet.
Apr 24th, 2015
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