Find, approximate, the area under f(x)=3x^2+6x +3 [-3,1] using the given partitions?
a) 4 upper sum rectangles
b) 2 midpoint rectangles
c) 2 trapezoids
d) 256 right sided rectangles

Question

Find, approximate, the area under f(x)=3x^2+6x +3 [-3,1] using the given partitions:
a) 4 upper sum rectangles
b) 2 midpoint rectangles
c) 2 trapezoids
d) 256 right sided rectangles

Luke Alvarez · Accepted Answer

Which ones are you having trouble with?

graph{3x^2+6x+3 [-14.44, 14.03, -0.74, 13.5]} (a) #12(1)+3(1)+3(1)+12(1) = 30# (b) #3(2)+3(2) = 12# (c) #1/2(12+0)(2) + 1/2(0+12)(2) = 12+12=24# (d) #sum_(i=1)^256 (3(-3+i/64)^2+6(-3+i/64) + 3) = (32,769)/(2,048)#

William Abdalla · Answer

undefined To approximate the area under the curve $f(x) = 3x^2 + 6x + 3$ over the interval $[-3, 1]$ using the provided partitions:

a) For 4 upper sum rectangles:
   Divide the interval $[-3, 1]$ into 4 equal subintervals.
   Calculate the width of each subinterval.
   For each subinterval, evaluate the function at the right endpoint and multiply it by the width of the subinterval.
   Sum up these products to approximate the area.

b) For 2 midpoint rectangles:
   Divide the interval $[-3, 1]$ into 2 equal subintervals.
   Calculate the midpoint of each subinterval.
   Evaluate the function at each midpoint and multiply it by the width of the subinterval.
   Sum up these products to approximate the area.

c) For 2 trapezoids:
   Divide the interval $[-3, 1]$ into 2 equal subintervals.
   Calculate the function value at both endpoints of each subinterval.
   Use the trapezoidal area formula for each subinterval.
   Sum up these areas to approximate the total area.

d) For 256 right-sided rectangles:
   Divide the interval $[-3, 1]$ into 256 equal subintervals.
   Calculate the width of each subinterval.
   For each subinterval, evaluate the function at the right endpoint and multiply it by the width of the subinterval.
   Sum up these products to approximate the area.

Perform the calculations according to the methods described above to obtain the respective approximations for the area under the curve.

Find, approximate, the area under f(x)=3x^2+6x +3 [-3,1] using the given partitions? a) 4 upper sum rectangles b) 2 midpoint rectangles c) 2 trapezoids d) 256 right sided rectangles

Find, approximate, the area under f(x)=3x^2+6x +3 [-3,1] using the given partitions: a) 4 upper sum rectangles b) 2 midpoint rectangles c) 2 trapezoids d) 256 right sided rectangles

Find, approximate, the area under f(x)=3x^2+6x +3 [-3,1] using the given partitions:
a) 4 upper sum rectangles
b) 2 midpoint rectangles
c) 2 trapezoids
d) 256 right sided rectangles