BvU said:
OmniNewton said:Homework Statement
Let A be the area of the region that lies under the graph of f(x) = 2x 2 + 5 between x = 0 and x = 4. Find an expression for A using n rectangles. Then evaluate this expression.
Homework Equations
Answer is 188/3
h= (4/n)
The Attempt at a Solution
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The problem I am having is I do not know how to simplify this limit further after factoring out the h.
BvU said:Well, the book answer helps you out, Ray
A definite integral by definition is a mathematical concept used to find the exact area under a curve. It is represented by ∫ and has a lower and upper limit, also known as the limits of integration.
To calculate a definite integral by definition, you must first divide the area under the curve into smaller rectangles or trapezoids. Then, you find the area of each shape and add them together to get an approximation of the total area. As the number of shapes increases, the approximation gets closer to the exact value.
A definite integral by definition involves using geometric shapes to approximate the area under a curve, while a definite integral by the Fundamental Theorem of Calculus involves using the antiderivative of a function to find the exact area. The latter is typically faster and more accurate, but it can only be used for certain functions.
The limits of integration determine the range over which the integral is being calculated. They specify the starting and ending points on the x-axis, which ultimately affects the value of the integral. Changing the limits of integration can result in a different area under the curve.
Yes, a definite integral by definition can be negative. This occurs when the curve dips below the x-axis, resulting in a negative area. The negative value simply indicates that the area under the curve in that region is being subtracted from the total area.