It was known even before Newton and Leibniz that areas under curves can be found by inverting the "computation of derivatives" (drawing tangents). In explicit geometric form this "fundamental theorem of calculus" was derived by Newton's teacher Barrow, see Barrow's Fundamental Theorem by Wagner.
Newton and Leibniz developed explicit symbolic methods for computing derivatives (isolated from the general process of drawing tangents), and anti-derivatives. To the extent that they conceptualized integral as such, it was identified with the anti-derivatives (what we call the indefinite integral), but, of course, it was used as a tool for computing areas, among other things. Euler still thought of it this way in 18th century. The idea of the definite integral does not appear until Cauchy in the 19th, see Kallio's History of the Definite Integral.