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How much area lies under y equals x squared plus one, between x equals zero and x equals two?

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Because the curve stays above the x axis, the definite integral gives exactly that area.

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We need the integral from zero to two of x squared plus one.

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An antiderivative is x cubed over three, plus x.

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The Fundamental Theorem says the integral equals F of two minus F of zero.

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F of two is eight thirds plus two, and F of zero is zero.

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So the area is fourteen thirds, about four point six seven square units.

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To recap: find an antiderivative, evaluate it at the upper and lower limits, and subtract.