How do you use the trapezoidal rule with n=6 to approximate the area between the curve #f(x)=x^2-9# from -3 to 3?

Question

Charles Arocha · Accepted Answer

The area (between the x-axis and) #f(x)=x^2-9# from #-3# to #3#
is approximately #-55# square units (the negative indicates the area is below the x-axis)

Given #f(x)=x^2-9# #{: (color(white)("X")x, color(white)("XXX"), f(x)), (-3, color(white)("XXX"), color(white)("X")0), (-2, color(white)("XXX"),-5), (-1, color(white)("XXX"), -8), ( color(white)("X")0, color(white)("XXX"), -9), (color(white)("X") 1, color(white)("XXX"), -8), (color(white)("X") 2, color(white)("XXX"), -5), ( color(white)("X")3, color(white)("XXX"), color(white)("X")0) :}# Since the width of each trapezoid is 1, the area will be equal to the average height. #{: ("trapezoid", color(white)("XX")"from x =", color(white)("XX")"to x=", color(white)("XX")"average height", color(white)("XX")"area" ), (" ", color(white)("XX")-3, color(white)("XX")-2, color(white)("XX")(0+(-5))/2=-2.5, color(white)("XX")-2.5 ), (" ", color(white)("XX")-2, color(white)("XX")-1, color(white)("XX")((-5)+(-8))/2=-6.5, color(white)("XX")-6.5 ), (" ", color(white)("XX")-1, color(white)("XXX")0, color(white)("XX")((-8)+(-9))/2=-8.5, color(white)("XX")-8.5 ), (" ", color(white)("XXX")0, color(white)("XXX")1, color(white)("XX")((-9)+(-8))/2=-8.5, color(white)("XX")-8.5 ), (" ", color(white)("XXX")1, color(white)("XXX")2, color(white)("XX")((-8)+(-5))/2=-6.5, color(white)("XX")-6.5 ), (" ", color(white)("XXX")2, color(white)("XXX")3, color(white)("XX")((-5)+0)/2=-2.5, color(white)("XX")-2.5 ) :}# Total area of trapezoids #= (-2.5)+(-6.5)+(-8.5)+(-8.5)+(-6.5)+(-2.5)# #= -55#

Isaiah Allen · Answer

undefined To use the trapezoidal rule with n=6 to approximate the area between the curve f(x)=x^2-9 from -3 to 3, follow these steps:

1. Divide the interval [-3, 3] into 6 equal subintervals.
2. Calculate the width of each subinterval, which is (b - a) / n, where a = -3, b = 3, and n = 6.
3. Evaluate the function f(x) = x^2 - 9 at the endpoints of each subinterval.
4. Use the trapezoidal rule formula for each subinterval: Δx * (f(x_i) + f(x_i+1)) / 2, where Δx is the width of the subinterval, x_i and x_i+1 are the endpoints, and f(x_i) and f(x_i+1) are the function values at those endpoints.
5. Sum up the areas obtained from each subinterval to get the approximate area under the curve.

Using this process, you would find the areas of the trapezoids for each subinterval and sum them up to approximate the total area under the curve.