# Calculus Worksheet

*label*Mathematics

*timer*Asked: Feb 2nd, 2019

*account_balance_wallet*$20

### Question Description

Please see the attached worksheet. It has four calculus problems involving integrals. There are some links provided in the document that may help in solving the problems.

## Tutor Answer

Here is the assignment ;)

1

Calculus worksheet

2

Answer:

a)

𝑅𝑛 = 10 = 1/2 = 0.5

Left Reimann sum = 0. 5( 𝑓(0) + 𝑓(0.5) + 𝑓(1.0) + 𝑓(1.5) + 𝑓(2.0) + 𝑓(2.5))

= 0.5( 3 + 4 + 5 + 6 + 7 + 8)

= 0.5 ∗ 33 = 𝟏𝟔. 𝟓

𝑛 = 100

Right Reimann sum = 0.5( 𝑓(0.5) + 𝑓(1.0) + 𝑓(1.5) + 𝑓(2.0) + 𝑓(2.5) + 𝑓(3.0))

= 0.5( 4 + 5 + 6 + 7 + 8 + 9)

𝐿𝑛 = 0.5 ∗ 39 = 𝟏𝟗. 𝟓

𝑛 = 1024

Exact area under curve = integral (1024 < 3) (from 0->3)

= [𝑥^2 + 3𝑥] (𝑓𝑟𝑜𝑚 0−> 3)

= [ 9 + 9 − 0 − 0]

𝑍𝑛 = 𝟏𝟖

b)

1

(b) Based on your calculations, is ∫0 𝑓(𝑥) dx well defined? Explain your reasoning. If so,

what does it equal?

1

This integral value in which applied of ∫0 𝑓(𝑥) is not well defined since here order of

integration only changed I.e in first part integration is done ln x to x as done on part a. Hence, in

x has the order of limits is changed. This indicates that both are reverse of each other but their

value remains same.

The region of the integration is bounded by 𝑥 = 𝑒 𝑦 = 𝑦 = ln x, x = 0, y = 0 and y =

ln1. Knowing that according to Boman & Rgers, when y=0, x=1(2014).

𝑦 → 0 𝑡𝑜 ln 𝑥

𝑥 → 1 𝑡𝑜 2

3

1

Then I = ∫0 𝑓(𝑥, )𝑑𝑥

𝟏

Hence, ∫𝟎 ...

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