#### To determine

a)

**To find:**

An expression for the area under the curve

#### Answer

*limn→∞∑i=1ni3n4*

#### Explanation

**1) Concept:**

The area *A* of the region *S* that lies under the graph of the continuous function *f* is

*A=limn→∞∑i=1nfxi∆x*

*∆x=b-an, xi=a+i∆x*

**2) Given:**

*y=x3, 0≤x≤1*

3) **Calculation:**

It is given that the curve

*y=x3, 0≤x≤1*

Here *a=0, b=1*

*∆x=b-an=1-0n=1n*

*∆x=1n*

*xi=a+i∆x*

*=0+i·1n*

*=in*

*xi=in*

Therefore, the area under the curve on *0, 1* is given by

*A=limn→∞∑i=1nfxi∆x*

Substitute the known values.

*=limn→∞∑i=1nfin·1n*

*=limn→∞∑i=1nin3·1n*

*=limn→∞∑i=1ni3n4*

**Conclusion:**

The expression for the area under the curve is

*A=limn→∞∑i=1ni3n4*

#### To determine

**b)**

**To evaluate:**

*limn→∞∑i=1ni3n4*

#### Answer

*14*

#### Explanation

**1) Given:**

*13+23+33+…+n3=nn+122*

2) **Calculation:**

From part a), we have

*limn→∞∑i=1ni3n4=limn→∞1n4∑i=1ni3*

*=limn→∞1n413+23+33+…+n3*

Use the given property.

*=limn→∞1n4nn+122*

*=limn→∞1n4n2n+124*

*=limn→∞n+124n2*

*=limn→∞14·1+1n2*

Simplify the limit.

*=14·1+02*

*=14*

**Conclusion:**

The area under the curve is

*limn→∞∑i=1ni3n4=14*