Evaluate the integral by interpreting it in terms of areas. integral: ∫−30(1+9−x2)dx

Mary Buchanan

Mary Buchanan

Answered

2021-12-19

Evaluate the integral by interpreting it in terms of areas. integral:
30(1+9x2)dx

Answer & Explanation

lenkiklisg7

lenkiklisg7

Expert

2021-12-20Added 29 answers

Step 1
First we break the integral into two parts:
30(1+9x2)dx=301dx+309x2dx
The first term 301dx represents area of rectangle with base b=3 and height h=1
301dx=b×h=3
309x2dx represent area of quarter circle with radius r=3
309x2dx=14π×32=9π4
30(1+9x2)dx=301dx+309x2dx
30(1+9x2)dx=3+9π4
Lakisha Archer

Lakisha Archer

Expert

2021-12-21Added 39 answers

Step 1
The given integral is
I=30(1+9xx)dx
I=301×x+309x2dx
I=A1+A2
we can plot area A1 as follows

we can plot area A2 as follows

Hence we can calculate the integral as
I=W×H+πr24
I=3×1+π(3)24
I=3+9π4
nick1337

nick1337

Expert

2021-12-28Added 573 answers

The expression you are having trouble with seems to be incorrect.
You have
f(x)=1+9x2
Since
a=3 and b=0
you have
Δx=3n and xk=3+3kn
Applying your definition:
f(xk)=1+9(3+3kn)2
=1+18kn9k2n2
=1+3n2knk2
From this, we see that
f(xk)Δx=3n(1+3n2knk2)
=3n+9n22knk2
Then the final expression should be
limn[k=1n(3n+9n22knk2)]

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