If you are learning about integration, there are some excellent online resources to help you. You can of course check your work with WolframAlpha and note that if you ask for a definite integral WolframAlpha will also return a visual representation of the integral, illustrating the area found. This is particularly helpful where parts of the curve are above the x axis and parts below. (You can see further calculus examples as part of Slideshow 4 on the WolframAlpha page).
Using the Integral Machine from The Center for technology and Teacher Education, University of Virginia, you can study approximate methods of integration such as the Trapezium Rule, try increasing the number of strips used and note how the area found using the Trapezium Rule (called Trapezoid here) gets closer to the actual area. You can also see from the shape of the curve why the trapezium rule can give an over or underestimate. Experiment with the four functions given.
If you use the Numerical integration utility and grapher from zweigmedia’s Finite mathematics and Applied Calculus you can choose your own function, then compare the actual value of the integral with that found by various approximate methods.