Find the area under the curve defined by $$ f(x) = 2x + 1 $$ between the points $$ x = 1 $$ and $$ x = 3 $$.

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Calculate the integral $$ \int_{1}^{3} (2x+1) \,dx $$.
Evaluate the integral by following steps:
- step0: Evaluate using formulas and rules:
  $$\int_{1}^{3} 2x+1 dx$$
- step1: Evaluate the integral:
  $$\int 2x+1 dx$$
- step2: Use properties of integrals:
  $$\int 2x dx+\int 1 dx$$
- step3: Evaluate the integral:
  $$x^{2}+\int 1 dx$$
- step4: Evaluate the integral:
  $$x^{2}+x$$
- step5: Return the limits:
  $$\left(x^{2}+x\right)\bigg |_{1}^{3}$$
- step6: Calculate the value:
  $$10$$
The area under the curve defined by $$ f(x) = 2x + 1 $$ between the points $$ x = 1 $$ and $$ x = 3 $$ is 10.
The area under the curve is 10.

Find the area under the curve defined by between the points and .

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