CALCU-Question 1The figure above shows the curve w

Question 1 The figure above shows the curve with equation

y=x^{2}-4 x+10, x \in \mathbb{R} .

Find the area of the region, bounded by the curve the coordinate axes and the straight line with equation

x=3

.

Generated Graph

Solution by Steps

step 1

To find the area of the region bounded by the curve

y = x^2 - 4x + 10

, the coordinate axes, and the line

x = 3

, we need to evaluate the definite integral of the function from

x = 0

to

x = 3

step 2

The definite integral of the function

y = x^2 - 4x + 10

from

x = 0

to

x = 3

is given by

\int_0^3 (x^2 - 4x + 10) \, dx

step 3

To compute the integral, we find the antiderivative of

x^2 - 4x + 10

, which is

\frac{x^3}{3} - 2x^2 + 10x

step 4

We then evaluate this antiderivative from

x = 0

to

x = 3

. The result is

\left[\frac{x^3}{3} - 2x^2 + 10x\right]_0^3

step 5

Plugging in the upper limit of integration, we get

\frac{3^3}{3} - 2(3)^2 + 10(3)

step 6

Plugging in the lower limit of integration, we get

\frac{0^3}{3} - 2(0)^2 + 10(0)

, which is simply 0

step 7

Subtracting the lower limit evaluation from the upper limit evaluation, we get

\left(\frac{27}{3} - 2(9) + 30\right) - 0

step 8

Simplifying the expression, we find

9 - 18 + 30 = 21

Answer

The area of the region is 21 square units.

Key Concept

Definite Integral as Area

Explanation

The definite integral of a function between two limits gives the area under the curve of the function and above the x-axis, between these two limits.

list all the specifications of international advanced level of math

AskSia