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tino4ka555 [31]
3 years ago
7

How to evaluate definite integral without the fundamental theorem of calculus?

Mathematics
1 answer:
AURORKA [14]3 years ago
7 0
If we can find the antiderivative function
F
(
x
)
of the integrand
f
(
x
)
, then the definite integral
∫
b
a
f
(
x
)
d
x
can be determined by
F
(
b
)
−
F
(
a
)
provided that
f
(
x
)
is continuous.
We are usually given continuous functions, but if you want to be rigorous in your solutions, you should state that
f
(
x
)
is continuous and why.
FTC part 2 is a very powerful statement. Recall in the previous chapters, the definite integral was calculated from areas under the curve using Riemann sums. FTC part 2 just throws that all away. We just have to find the antiderivative and evaluate at the bounds! This is a lot less work.
For most students, the proof does give any intuition of why this works or is true. But let's look at
s
(
t
)
=
∫
b
a
v
(
t
)
d
t
. We know that integrating the velocity function gives us a position function. So taking
s
(
b
)
−
s
(
a
)
results in a displacement.
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c

Step-by-step explanation:

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Molly was on a long 136 mile road trip. The first part of the trip there was lots of traffic, she only averaged 16 mph. The seco
Mazyrski [523]

Answer:

In traffic, she drove for 3 hours

and After the traffic cleared she drove for 2 hours.

Explanation:

Given that the road trip was 136 miles;

d=136

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v_1=16

The second part of the trip there was no traffic so she could drive 44 mph;

v_2=44

She traveled for a total of 5 hours;

t=5

let x represent the time in traffic when she traveled at 16 mph

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the time the traffic is clear would be;

t_2=t-t_1=5-x

Recall that distance equals speed multiply by time;

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solving for x;

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7 0
1 year ago
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evablogger [386]

Answer: 5


Step-by-step explanation:

 1. You have the following expression given in the problem above:

\frac{\sqrt{97.5}}{1.96}

2. You can estimate the result by rounding the numerator and the denominator.

3. As you can see, you can round up 97.5 to 100.

4. Then, you can round up 1.96 to 2.

5. Therefore, you have:

\frac{\sqrt{100}}{2}

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Answer:

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6 * 2.2 ≥ x

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