Exam (elaborations)
The Definite Integral and the Fundamental Theorem of Calculus solved questions
- Course
- Institution
The Definite Integral and the Fundamental Theorem of Calculus solved questions
[Show more]
Seller
Follow
jureloqoo
Content preview
CHAPTER 20The Definite Integral and the
Fundamental Theorem of Calculus
20.1 Evaluate 4 dx by the direct (Riemann) definition of the integral.
Let 2 = x0 <xl < • • • < xn_i < xn =5 be any partition of [2,5], and let A,JC = x. —*,•_,. Then an
approximating sum for 4 dx is
= 4(jt,, - x0) = 4(5 - 2) = 4 • 3 = 12.
Hence, the integral, which is approximated arbitrarily closely by the approximating sums, must be 12.
20.2 Calculate 5x2dx by the direct definition of the integral.
Divide [0,1] into n equal subintervals, each of length A,* = l//i. In the /th subinterval, choose x* to be the
right endpoint i/n. Then the approximating sum is
As we make the subdivision finer by letting n —» +», the approximating sum approaches | • 1 • 2 =which
f,
is the value of the integral.
20.3 Prove the formula that was used in the solution of Problem 20.2.
Use induction with respect to n. For n = l. the sum consists of one term (I) 2 = 1. The right side
is ( l - 2 - 3 ) / 6 = l . Now assume that the formula holds for a given positive integer n. We must prove it
for n + I. Adding (n + I) 2 to both sides of the formula we have
which is the case of the formula for n + 1.
20.4 Prove the formula 1+ 2+ • • •+ n =
Let S = 1 + 2 + • • • + (n - 1) + n. Then we also can write 5 = n + (n - 1) + • • • + 2 + 1. If we add
these two equations column by column, we see that 25 is equal to the number n + I added to itself n times.
Thus, 25 = «(« + !), S = n ( n + l)/2.
20.5 Show that by the direct definition of the integral.
Divide the interval [0, b\ into n equal subintervals of length bin, by the points 0 = x0 < bin <2bln<- • • <
nbln = x = b. In the ith subinterval choose x* to be the right-hand endpoint Ibln. Then an approximating
sum is As «—»+«>, the approximating sum approaches
b 12, which is, therefore, the value of the integral.
152
, THE DEFINITE INTEGRAL AND THE FUNDAMENTAL THEOREM OF CALCULUS 153
20.6 Evaluate
20.7 Evaluate
20.8 For the function / graphed in Fig. 20-1, express f(x) dx in terms of the areas Al, A2, and A3.
Fig. 20-1
I The integral is equal to the sum of the areas above the *-axis and under the graph, minus the sum of the areas
under the x-axis and above the graph. Hence, J05 f(x) dx = A2 - Al - A3.
In Problems 20.9-20.14, use the fundamental theorem of calculus to compute the given definite integral.
20.9 (3x2 - 2x + 1) dx.
(3x2 -2x + l)dx = x3-x2 + x. (We omit the arbitrary constant in all such cases.) So (3x2 -2x +
l ) ^ = ( ^ 3 - x 2 + ^)] 3 _ 1 = (3 3 -3 2 + 3)-[(-l) 3 -(-l) 2 + (-l)] = 21-(-3) = 24.
20.10 cos x dx.
cos x dx = sin x. Hence,
20.11 J7/3 sec2 x dx.
sec2 x dx = tan x. Hence,
20.12 x312 dx
x312 dx = f x5'2. Hence,
20.1
dx = $ (2x'l/2 -x)dx = 4xl/2 - \x2. Hence,
The benefits of buying summaries with Stuvia:
Guaranteed quality through customer reviews
Stuvia customers have reviewed more than 700,000 summaries. This how you know that you are buying the best documents.
Quick and easy check-out
You can quickly pay through credit card or Stuvia-credit for the summaries. There is no membership needed.
Focus on what matters
Your fellow students write the study notes themselves, which is why the documents are always reliable and up-to-date. This ensures you quickly get to the core!
Frequently asked questions
What do I get when I buy this document?
You get a PDF, available immediately after your purchase. The purchased document is accessible anytime, anywhere and indefinitely through your profile.
Satisfaction guarantee: how does it work?
Our satisfaction guarantee ensures that you always find a study document that suits you well. You fill out a form, and our customer service team takes care of the rest.
Who am I buying these notes from?
Stuvia is a marketplace, so you are not buying this document from us, but from seller jureloqoo. Stuvia facilitates payment to the seller.
Will I be stuck with a subscription?
No, you only buy these notes for $8.04. You're not tied to anything after your purchase.
Can Stuvia be trusted?
4.6 stars on Google & Trustpilot (+1000 reviews)
67474 documents were sold in the last 30 days
Founded in 2010, the go-to place to buy study notes for 14 years now