Use a numerical method to find the area enclosed by the graph of y = x² sin x, the x axis and the lines x = 1 and x = 2.6
Use an interval of 0.4
We can approximate via trapezoid rule for the function is continous.
In order to use the trapezoid rule tou need to find the number of intervals.
In this case the length is 2.6-1=1.6
Then divided by the length of each one 1.6/.4=4
Thus, a use of 4 intervals.
By the trapezoid rule we need to find,
f(1),f(1.4),f(1.8),f(2.2),f(2.6)
We note that,
f(1)=(1)^2*sin(1)=0.84
f(1.4)=(1.4)^2*sin(1.4)=1.93
f(1.8)=(1.8)^2*sin(1.8)=3.16
f(2.2)=(2.2)^2*sin(2.2)=3.91
f(2.6)=(2.6)^2*sin(2.6)=3.48
Now we double the inner terms (look at formula)
Thus,
f(1)=.84
2f(1.4)=3.86
2f(1.8)=7.32
2f(2.2)=7.82
f(2.6)=3.48
Add them together,
Result=23.32
Now multiply it by the length of each interval, (.4)
Thus,
9.33
Finally multiply by 1/2,
4.67
That is the approximate answer.
Hello, norivea!
Use a numerical method to approximate the area enclosed by:
the graph of y = x² sin x, the x axis and the lines x = 1 and x = 2.6
Use an interval of 0.4
There are many available methods.
I'll use left-endpoints.
. f(1) . = . .1²·sin(1) . . ≈ .0.8415
f(1.4) .= .1.4²·sin(1.4) .≈ .1.9315
f(1.8) .= .1.8²·sin(1.8) .≈ .3.1553
f(2.2) .= .2.2²·sin(2.2) .≈ .3.9131
f(2.4) .= .2.4²·sin(2.4) .≈ .3.8907
f(2.6) .= .2.6²·sin(2.6) .≈ .3.4845
Area .≈ .(0.8415 + 1.9315 + 3.1553 + 3.9131 + 3.8907 + 3.4845) x 0.4 .= .6.88676
When using the left end point method we dont use your last point f(2.6),
also all the increments should be equal.
So your solution should look like:
. f(1) . = . .1²·sin(1) . . ≈ .0.8415
f(1.4) .= .1.4²·sin(1.4) .≈ .1.9315
f(1.8) .= .1.8²·sin(1.8) .≈ .3.1553
f(2.2) .= .2.2²·sin(2.2) .≈ .3.9131
Area .≈ .(0.8415 + 1.9315 + 3.1553 + 3.9131) x 0.4 .=3.94
RonL
The actual value of the integral is of no relevance here, we have an
excercise in numerical integration with a specified step size. The
answer for any particular method is very unlikely to agree with the
answer obtained analyticaly.
Also 4.46 is closer to 4.53 than your answer of 4.67.
RonL