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- Area Under Curves and Volume of Revolving a Curve

# Area Under Curves and Volume of Revolving a Curve

# Area Under Curves and Volume of Revolving a Curve

**Definite Integrals via Numerical Methods**

This relates to definite integration via numerical methods.

Consider the algebraic expression given by:

_{}^{} _{}^{} _{}^{} _{}^{}

For the purpose of numerical computation, the area under the curve between the limits and can be computed by the Limit Definition of a Definite Integral.

Here is some background about **areas and volume computation**.

Using equal subintervals of length , you need to:

Evaluate the area bounded by a given polynomial function of the kind described above, between the given limits of and .

Evaluate the volume of the solid obtained by revolving this polynomial curve around the -axis.

A relative error margin of will be tolerated.

**Input Format**

The first line contains integers separated by spaces, which are the values of _{}_{}_{}.

The second line contains integers separated by spaces, which are the values of _{}_{}_{}.

The third line contains two space separated integers, and , the lower and upper range limits in which the integration needs to be performed, respectively.

**Constraints**

**Output Format**

The first line should contain the area between the curve and the -axis, bound between the specified limits.

The second line should contain the volume of the solid obtained by rotating the curve around the -axis, between the specified limits.

**Sample Input**

```
1 2 3 4 5
6 7 8 9 10
1 4
```

**Explanation**

The algebraic expression represented by:

^{}^{}^{}^{}^{}

We need to find the area of the curve enclosed under this curve, between the limits and . We also need to find the volume of the solid formed by revolving this curve around the -axis between the limits and .

**Sample Output**

```
2435300.3
26172951168940.8
```

**Scoring**

All test cases are weighted equally. You need to clear all the tests in a test case.