he integral represents the area between the two curves f(x)=x+4𝑓(𝑥)=𝑥+4 and g(x)=4−x2𝑔(𝑥)=4−𝑥2 and between left and right boundaries for x as shown in the Figure below.Geogebra plot. You can zoom with the mouse wheel and move the graph by clicking on the background and dragging.__________________________________________________a) Express the area as a definite integral:
Question
he integral represents the area between the two curves f(x)=x+4𝑓(𝑥)=𝑥+4 and g(x)=4−x2𝑔(𝑥)=4−𝑥2 and between left and right boundaries for x as shown in the Figure below.Geogebra plot. You can zoom with the mouse wheel and move the graph by clicking on the background and dragging.__________________________________________________a) Express the area as a definite integral:
Solution
To express the area between the two curves as a definite integral, we first need to find the points of intersection of the two curves. These points will serve as the limits of integration.
The two curves intersect when f(x) = g(x), so we solve the equation x + 4 = 4 - x^2 for x. This simplifies to x^2 + x - 4 = 0. Solving this quadratic equation gives us the roots x = -2 and x = 2.
The area A between the two curves from x = -2 to x = 2 is given by the definite integral of the absolute difference of the two functions from -2 to 2. Since g(x) is above f(x) in this interval, we subtract f(x) from g(x).
So, the area A is given by:
A = ∫ from -2 to 2 of (g(x) - f(x)) dx = ∫ from -2 to 2 of ((4 - x^2) - (x + 4)) dx = ∫ from -2 to 2 of (-x^2 - x) dx
This is the expression for the area as a definite integral.
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