3 INTEGRAL CALCULUS AND MODELLING questions Essay Example

For

In this case a = 0 b =64 f(x) =

= =

for each case of i

= for each i

Through expansion this equation is reduced to

-10) = 64

This gives n = 2.4

b) summation notations

Expression for upper Riemann sum

Expression for lower Riemann sum

c) Finding exact value of integral

= = 191.7

2 Evaluation of integrals

(B-A)x (-5A-B) =

Thus B-A = 1 and -5A –B = -2

Solving for A and B

A = B =1

=

In

= ( In — (In

b) = dx = dx

dx

Using u = cosx du = -sinxdx

dx

= +C

=

= —

c) dx = dx = In

Areas calculation

Curve Can be written as

While curve can also be written as

The required region given by

Areas bound by the y-axis and the line y =5 between coordinates 5, 25 and 5,625

= (5×625) – (5×25) =3125 – 125 =3000

Areas bound by the y-axis and the curve between coordinates 4, 16 and 5,25

A₂ = = = 83.33- 42.66 = 40.67

A₃ = = = 2500 — 44.7 = 2455.3

Required area = +A₂)-A₃ = (3000 +40.67) -2455.3 = 585.37

Calculation volume

The volume is the space swept by the area about the axis x= -1

=

= 11033 units