Until the invention of calculus, there was no general way to determine the area beneath any given curve. The integral of a function over an interval [a, b], known as the arc length of the curve or function, defines the area under the function over that interval. The process of determining the arc length of a curve (function) is called rectifying the curve. For any real-valued function f(x), where f(x) and f′(x) are continuous over the interval [a, b], the arc length s of the curve is . Depending on the form of the function, ds has different forms: for rectangular coordinates, ; for polar coordinates, ; and for parametric equations, .
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Consider the integral from 0 to 10 of x^2*g'(x) dx where g has the following table:
(if x^2g(x) evaluted from 0 to 10 -minus integral from 0 to 10 2xg(x)dx represents the first step using integration by parts)
Approximate the integral. Use the right hand Riemann sum with n=5 to approximate the second term?•
Consider the graph of y = e^x.
(a) Find the equation of the graph that results from reflecting about the line y = 7.
y = ?
(b) Find the equation of the graph that results from reflecting
about the line x = 3.
y = ?
a)find the unit vectors that are parallel to the tangent line to the curve y=2 sin x at the point (pi/6,1).
b)find the unit vectors that are perpendicular to the tangentline.•
Consider the IVP for the function given by
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