#### To determine

**a)**

**To find:**

The number a such that the line x=a bisects the area under the curve y=1x2, 1≤x≤4

#### Answer

85

#### Explanation

**1) Concept:**

Bisect means the cut the area in half. So, area above the bisecting line is the same as the area below the bisecting line.

**2) Given:**

The curves y=1x2, 1≤x≤4

**3) Formula:**

∫xn dx=xn+1n+1+C

4) **Calculation:**

To find the line that bisects the area under the curve.

Bisect means the cut the area in half. So, area above the bisecting line is the same as the area below the bisecting line.

Let the line x=a bisect the area under the curve y=1x2, 1≤x≤4

Therefore, ∫1a1x2dx=∫a41x2dx

Solve integration.

-1x1a=-1xa4

Substitute limits.

-1a+1=-14-(-1a)

Simplify.

1-1a=1a-14

Add 14 on both sides.

1-1a+14=1a-14+14

Simplify.

54-1a=1a

Add 1a to both the sides.

54-1a+1a=1a+1a

Simplify.

54=2a

Cross multiply.

5a=8

Divide by 5, and simplify.

a=85

Therefore, the number 85 is such that the line x=85 bisects the area under the curve y=1x2

**Conclusion:**

The number 85 is such that the line x=85 bisects the area under the curve y=1x2

#### To determine

**b)**

**To find:**

The number b such that the line y=b bisects the area in part a)

#### Answer

≈ 0.1503

#### Explanation

**1) Concept:**

Bisect means the cut the area in half.

2) **Calculation:**

The area under the curve y=1x2 from x=1 to 4 is given by,∫141x2dx=∫14x-2dx

Using power rule of integration,

=x-1-114

=-1x14

Plug limits,

=-14+1

=34

Therefore,

Area=34

To find the number b such that the line y=b bisects the curve x=1y and not the line x=4.

Since, the area under the line y=142 from x=1 to x=4 is only 316 which is less than half of 34

Now, choose b such that upper area in diagram is half of the total area under the curve y=1x2

From x=1 to 4

∫b11y-1dy=1234

Simplify.

2y-yb1=38

Plug limits.

21-1-2b-b=38

Simplify bracket.

1-2b+b=38

b-2b+58=0

Letting c=b,

Therefore,

c2-2c+58=0

Simplify,

8c2-16c+5=0

By using quadratic formula,

c=16±4256-16016

=1±64

But, c=b<1

⇒c=1-64

b=c2=1-642=1+38-62=1811-46≈0.1503

The value b≈2.599 is too high since it is higher than the line in part a.

Therefore, b≈ 0.1502.

Therefore, the number b≈ 0.1503 is such that the line x=0.1503 bisects the area in

part a).

**Conclusion:**

Therefore,

The number b≈ 0.1503 is such that the line x=0.1503 bisects the area in part a).